ARITHMETICAL BOOKS.

LONDON:
PRINTED BY ROBSON, LEVRY, AND FRANKLYN,
Great New Street, Fetter Lane.

LONDON
TAYLOR AND WALTON
BOOKSELLERS AND PUBLISHERS
TO UNIVERSITY COLLEGE
28 UPPER GOWER STREET
1847

[Pg i]

MY DEAR SIR,

It never entered into my head till now to adorn the front of any book of mine with an eminent name: and the reason I take to be, that I have hitherto never chanced to write a separate work⁠[1] upon any subject with which the name of one individual was especially associated in the minds of those who study it. But you are the only Englishman now living who is known, by the proof of publication, to have investigated both the scientific and bibliographical history of Arithmetic: and this compliment, be the same worth more or less, is your due, and would have been, though my knowledge of you had been confined to your writings. And it is the more cordially paid from the remembrance of nearly a quarter of a century of personal acquaintance, and of many acts of friendship on your part.

I have rather grown than made this catalogue. It never occurred to me to publish on the subject, till I found, on a casual review of what I had collected, that I could furnish from my own books a more extensive list than Murhard, Scheibel, Heilbronner, or any mathematical bibliographer of my acquaintance, has described from his own inspection. Knowing, from sufficient experience, the general inaccuracy and incompleteness of scientific lists, I therefore determined to do what I could towards the correction of both, by [Pg ii] describing as many works as I could manage to see. From the Royal Society’s library, the stock of Mr. Maynard the mathematical bookseller, and my own collections, with a few from the British Museum and the libraries of private friends, including three or four of great rarity from yourself, I have accordingly compiled the present catalogue. I have also given in the Index, in addition to the names of the authors whom I have examined, those of all whom I could find recorded as having written on the subject of Arithmetic, whether as teachers, historians, or compilers of special tables for aid in the main operations, independently of logarithms and trigonometry.

A great number of persons are employed in teaching arithmetic in the United Kingdom. In publishing this work, I have the hope of placing before many of them more materials for the prevention of inaccurate knowledge of the literature of their science than they have hitherto been able to command, without both expense and research. Your History, unfortunately for them, is locked up in the valuable, but bulky and costly, Encyclopædia for which it was written. I may have the gratification of knowing that some, at least, of the class to which I belong, have been led by my catalogue to make that comparison of the minds of different ages which is one of the most valuable of disciplines, and without which the man of science is not the man of knowledge.

The most worthless book of a bygone day is a record worthy of preservation. Like a telescopic star, its obscurity may render it unavailable for most purposes; but it serves, in hands which know how to use it, to determine the places of more important bodies.

The effect of this work which would please me best, would be, that the professed bibliographer should find it too arithmetical, and that the student of the history of the science should find it too bibliographical. I might certainly have entered more into the methods of the several works, with advantage to the reader. But I could not attempt to write the complete annals of arithmetic; this would require still more books: neither could I, without losing sight of my plan altogether, combine the information here presented with that derived from other sources. That plan has been, to attempt some rectification of the numerous inaccuracies of existing catalogues, by recording only what I have seen myself. And it is a sufficient justification of the course I have taken, that I have produced in this way a larger catalogue than yet exists among those devoted, in whole or in part, to this particular subject. [Pg iii]

None but those who have confronted the existing lists with the works they profess to describe, know how inaccurate the former are; and none but those who have tried to make a catalogue know how difficult it is to attain common correctness. There is now a prospect of this country possessing in time such a record of books as can be safely consulted in aid of the history of literature,—I refer to the intended catalogue of the library in the British Museum. I, for one, can only hope that the chance will not be lost by any attempt to expedite its formation, in deference to the opinion of those who either are not aware how bad existing lists are, or are willing to take more than a chance of having nothing better. If, through negligence or fear on the part of those who have really compared book-lists and books, the expression of public feeling which any primâ facie case against public officers so easily obtains, should succeed in hurrying the execution of this national undertaking, the result will be one more of those magazines from which non-existing books take their origin, and existing ones are consigned to oblivion by incorrect description. However extensive the demand for spoiled paper may be, it should be remembered that the supply is immense, and that there is no need to insist on Parliament, at least, furnishing a larger contingent than it is obliged to do already.

I make no apology for troubling you to read this diatribe: it is your affair, and mine, and that of every one who believes accuracy to be an essential characteristic of useful knowledge.

I remain, 
My dear Sir, 
Yours sincerely, 
A. DE MORGAN.

University College, London,
April 29, 1847.

[Pg iv]

⁂ In consequence of a little change in the plan of this work, made after the printing was commenced, there may be one or two places in which the reader must consult the Introduction, where he is told to consult the Index; and the Additions, where he is told to consult the Introduction.

At the end of this work will be found an Index containing, besides the names of the authors mentioned in the Catalogue, who may be known by the paginal figures opposite, every name which I have found any where as belonging to a writer on arithmetic before 1800. Since that period I have not swelled the list from mere catalogues, but have contented myself with what I had either seen, or learnt from some other source. For the existence of such writers, or of their works, of course I do not vouch, except in those cases in which the work is found in the body of the Catalogue. Of those who stand unpaged in the index, all I can say is, that they are taken from many sources, and that there is in each case plenty of reason to make inquiry for their writings, and strong presumption that their works are still to be found.

A word with an asterisk before it is the name of a writer who is referred to by Dr. Peacock. The great apparent preponderance of German names arises partly from the number of works written on the subject in Germany being very considerable, partly from the bibliographical catalogues of that country being more full than those of any other, and partly from the names which seem to be German, really including Danes, Dutch, Belgians, and some Swiss. I am far from thinking that this list contains even the third part of the names of those who have really written on the subject. I have not been particular in searching for any thing after 1750, though I have not refused what came in my way. I might have very much increased the list of recent Germans, from Rogg’s Bibliotheca Selecta (Tubingen, 1830). Looking at the various countries which enjoyed the art of printing from 1500 downwards, I have an impression, from all that I have gathered, which would lead me to suppose that the number of works on arithmetic published in Latin, [Pg vi] French, German, Dutch, Italian, Spanish, and English, up to the middle of last century, cannot be less than three thousand, which gives to each language less than an average of one a year. Few of these would seem to fall within the province of the historian. Dr. Peacock refers to about a hundred and fifty. Unfortunately, history must of necessity be written mostly upon those works which, by being in advance of their age, have therefore become well known. It ought to be otherwise, but it cannot be, without better preservation and classification of the minor works which people actually use, and from which the great mass of those who study take their habits and opinions. The Principia of Newton is, if we believe the title-page, a work of the seventeenth century: but the account of the effect which it produced on science belongs to the eighteenth. It was not till many years after the publication of the Principia, that its predecessor in doctrine, the great work of Copernicus, produced its full effect upon general thought and habit. Nor have we any reason to suppose that it could have been otherwise. The great exceptions will always bear, perhaps, as large a ratio to the average power as ever they did: it is as likely as not, that if the intelligence of the sixteenth century had been sufficient to verify and receive the opinion of Copernicus at once, some predecessor of his might have been Copernicus, and he or another of his day might have been Newton.

It is then essential to true history, that the minor and secondary phenomena of the progress of mind should be more carefully examined than they have been. We must distinguish between the progress of possibilities and that of actual occurrences. Our written annals shew us too much what might have been, and too little what was: they give some words to the slow reception of an improvement, and more sentences to the account of the one man who was able to make it before the world at large could appreciate it.

The public is beginning to demand that civil history shall contain something more than an account of how great generals fought, great orators spoke, and great kings rewarded both for serving their turn. The progress of nations might well be described, for the most part, with much less mention of any of the three: but the parallel does not hold of knowledge. Copernicus and Newton would fill a large space, though the history were written down to that of every individual who ever opened a book: but it seems to me that they, and their peers, are made to fill all the space. Nor will it be otherwise until the [Pg vii] historian has at his command a readier access to second and third rate works in large numbers; so that he may write upon effects as well as causes.

This list contains upwards of fifteen hundred names, of which a few may be duplicates of others, arising from the wrong spelling of my authorities.⁠[2] But these must be much more than counterbalanced by the number of names which belong to two or more authors, but which only appear once in my list. Thus there are two Digges’s, three Riese’s, two named Le Gendre, Walker, Newton, Wallis, &c., and several Taylors, Butlers, &c., though each name is only mentioned once. There are several cases in which I have not ventured to strike out one of two names, though there is every reason to suppose one is a mistake for the other; as Caraldus and Cataldus, Cappaus and Cuppaus, Döhren and Dühren. But I have often been deceived in this way; and have more than once been obliged to re-insert names which I had struck out, supposing them to be only different spellings of names already in the list. There are several whom I have seen in more places than one, who are clearly Germans with names metamorphosed by wrong reading of the black letter, or use of the genitive for a nominative, or both. Thus Petzoldt has become Pekoldts, Seckgerwitz has been made Sectgerwik, Schultze has been made Schulken, and so on. These obvious mistakes of course have not been admitted. Moreover, several persons are, I suppose, down in the list under two names by which they are known. Thus, Fortius may be Ringelbergius (whose real name was Sterk), Blasius may be Pelacanus. But as I cannot undertake to assert that there is no Fortius except Ringelbergius, &c., I have let both names stand in the list. Again, next to Buckley comes Budæus. Many foreign writers, Heilbronner among the rest, have turned Buclæus into Budæus, so that in all probability the second of these names is a mistake for the first. And yet there may be also some Budæus who has written on arithmetic; though, having excluded writers on weights and measures only, I have not put down the author of the work De Asse.

I have had, in one or two instances, to throw away German authors, for a very obvious reason. The reader will not find the works of Anleitung, or Grundriss, or Rechenbuch in my list, which is more than can be said of every one which has preceded it. [Pg viii]

I have not attempted to translate the names of those who wrote in Latin, at a time when that language was the universal medium of communication. In every such case I consider that the Latin name is that which the author has left to posterity; and that the practice of retaining it is convenient, as marking, to a certain extent, the epoch of his writings, and as being the appellation by which his contemporaries and successors cite him. It is well to know that Copernicus, Dasypodius, Xylander, Regiomontanus, and Clavius were Zepernik, Rauchfuss, Holtzmann, Muller, and Schlüssel. But as the butchers’ bills of these eminent men are all lost, and their writings only remain, it is best to designate them by the name which they bear on the latter, rather than the former. In some cases, as in that of Regiomontanus, both names are frequently used: in others, where the Latin consists in a termination only, as Tonstallus for Tonstall, or Paciolus for Pacioli, it matters nothing which is used. It may happen that errors are introduced by returning to the vernacular in a wrong way. I should like to know how it is shewn that Orontius Fineus was Oronce Finée: or in what respect this reading is more finically correct than Horonce Phine, which has great antiquity in its favour? In the case of Vieta, Viète is certainly wrong: he was called by his contemporaries Viet (which I suspect to be derived from the Latin form) and de Viette, never Viète.

It may also be asked, why the unlatinizing process should, for confusion’s sake, be practised by the learned only; when it is pretty certain that the world at large will never reconvert Melancthon into Schwartzerd, or Confucius into Kaen-foo-tzee. Neither will they restore to the Popes and other priests of the Roman Catholic Church the names under which they were born and educated.

In many cases it would be impossible to recover these last. For myself, I am well content with the name under which an author was known in literature to his contemporaries, and has been handed down to us, his successors. I know of no canon under which it is imperative to speak of a writer rather as his personal acquaintance than as his reader: and, so far as feeling of congruity is concerned, I think Alexander ab Alexandro looks better at the head of a Latin preface than Saunders Saunderson. Those who really wish to catch the tone of the middle ages, and shew themselves quite at home, should dwell on the Christian name, and make the surname a secondary distinction; should learn to think of Nicolas (who happened to be called Copernicus, or Zepernik, it should matter little which) and Christopher, to whom the calendar was entrusted. [Pg ix]

That this list must be very imperfect I am well aware, for I have been able to add many names to it which I never found in any catalogue. But it will not be useless. It may furnish a reason for preserving any work whatsoever which comes in the way of the reader of this book. If the name be in the list, a book should not be destroyed which has been somewhere catalogued and recognised as a portion of the existing materials for the history of science. But if the name be not in the list, it is obvious that there is some curiosity about a writer whose name is not in the most numerous catalogue of arithmetical authors that has ever been collected in any one place. And it is undeniable that every name must be in one or the other predicament.

I now come to the Catalogue which forms the body of this work. The defects, which every one who has examined the lists that are extant, knows to prevail, arise, in great measure, from the titles and descriptions of books being copied by those who had never seen the books themselves. This is not the worst; for a true copy of a true copy is a true copy: many of these accounts have finally become formal representations of informal titles, which, though not originally intended for more than sufficient indication of the book mentioned, have been used as if they had been full and accurate descriptions. There is a large number of works, not much distinguished in the history of science, each of which has, nevertheless, done its part in its day. The minor points of the same history depend much upon these books, which, being neither of typographical curiosity, nor of literary fame, are gradually finding their way either to the waste paper warehouse, or the public library. These two depositories are almost equally unfavourable to works of no note assuming their place in the annals of the knowledge to the progress of which they have contributed. Take the library of the British Museum, for instance, valuable and useful and accessible as it is: what chance has a work of being known to be there, merely because it is there? If it be wanted, it can be asked for; but to be wanted, it must be known. Nobody can rummage the library, except those officially employed there, who will only now and then have leisure to turn their opportunities to account in any independent literary undertaking. And it would perhaps be difficult to make any regulation, under which persons not belonging to the institution might have access to see what is there. Nor will the publication of the [Pg x] catalogue do much towards supplying the defect. Titles of books are but vague indicators of their contents; and a catalogue⁠[3] of half a million of entries, even if its contents could be guessed at by the titles of the books, is not made to be read through.

It would be something towards a complete collection of mathematical bibliography, if those who have occasion to examine old works, and take pleasure in doing it, would add each his quotum, in the shape of description of such works as he has actually seen, without any attempt to appear more learned than his opportunities have made him. This is what I have done in the descriptive catalogue of works on arithmetic, without selection, or other arrangement than order of date. The only reason for a work being in this list, is that it has come in my way: the only reason for one being out of it is, that it has not come in my way, at least at the time of compilation. Whenever any statement is made which is taken from any other writer, it will be put in brackets [], except when the statement itself cites an authority: though I have sometimes put the brackets even in the latter case. The only other mode of proceeding would be, to collect lists from authority, naming the sources of information. But, having found so many errors in these sources, when my opportunities have enabled me to bring them to the test, I did not feel inclined to be the tenth transmitter of inaccurate copies. My mistakes shall be of my own making, and it would not be easy to invent one which should want high precedent for its species.

The description of the books in the Catalogue is uniformly as follows. There are given:

1. The place at which the work was printed, in Italics, and generally in English.

2. The date of the title-page, or colophon, or, when both are wanting, of the preface, in words; but not quite at length. Thus instead of fifteen-hundred-and-eighty-eight will be found fifteen-eighty-eight. I am sure that dates will never be given correctly until this plan is universally adopted: for two reasons. First, the chance of error in printing is very much diminished: particularly the risk of a transposition of figures at the press. Secondly, the writer, who [Pg xi] is, on the whole, and one date with another, more to be feared than the printer, has time to be accurate. A glance at four numerals and four strokes with the pen, is too rapid a process for certainty: and those who think they can rely upon themselves to stop every time, and look at what they have done, will frequently find reason to wish they had been less confident. But Incidit in Scyllam, &c.: I had very nearly announced an edition of Cocker as being unmistakably the seventeen-hundred-and-twentieth (see page 56).

3. The author’s name. When an initial only is given, it is because the author has left no more, either in title or preface.

4. As much of the title as will certainly identify the book. In spelling, initial capitals, &c., I have generally followed the author closely. When there is any defect in this respect, I suppose it will be in those works which I had not any opportunity of re-examining while the sheets were in proof. Imitation of type I have not attempted. To have given the fall titles would have swelled my book too much: at certain periods the authors of elementary works were much given to write out descriptive chapters in their title-pages; scores of them would each have filled two or three pages of even the smaller type used in the Catalogue.

5. The form in which the work is printed; a matter which will require some explanation.

A folio, quarto, octavo, duodecimo, or smaller work, is now generally known by its size, though not always. In the folio the sheet of paper makes two leaves or four pages, in the quarto four leaves, in the octavo eight, in the duodecimo twelve, and so on. But even a publisher thinks more of size than of the folding of the sheet when he talks about octavo or quarto; and accordingly, when he folds a sheet of paper into six leaves, making what ought to be a sexto book, he calls it a duodecimo printed in half sheets, because such printing is always done with half-sized paper, or with half sheets, so as to give a duodecimo size. From a very early period it has been universal to distinguish the sheets by different letters called signatures. In the book now before the reader, which is a half-duodecimo (or what I call a duodecimo in threes), the first sheet which follows the prefatory matter, B, has B on the first leaf, and B 2 on the third; which is enough for the folder’s purpose. But in former times the signatures were generally carried on through half the sheet, and sometimes through the whole. Again, in modern times, no sheet ever goes into and forms part of another; that is, no leaf of any one sheet ever [Pg xii] lies between two leaves of another. But in the sixteenth century, and even later in Italy, it was common enough to print in quire-fashion. Imagine a common copybook, written through straightforward, and the string then cut: and suppose it then separates into four double leaves besides the cover. It would then have sixteen pages, the separate double leaves containing severally pages 1, 2, 15, 16; 3, 4, 13, 14; 5, 6, 11, 12; 7, 8, 9, 10. If a book were printed in this way, it would certainly be a folio, if the four double leaves of any one quire or gathering were each a separate sheet: and if the sheet were the usual size, it would give the common folio size. But if each gathering had the same letter on all its sheets, if the above for instance were marked A₁ on page 1, A₂ on page 3, A₃ on page 5, and A₄ on page 7; the book, when made up, would have all the appearance of a more recent octavo in its signatures. In order to give the size of a book, and at the same time to give the means of identifying the edition by its signatures, I have adopted the plan which gives the following rules:

[Pg xiii] By this means there is something as to size, and something as to signatures, in every description. But whether any book which I call octavo in twos, for instance, really was printed on whole sheets, or on half sheets, that is, really was a small quarto, or a divided octavo, is more than I can in any case undertake to say. All I know is, that with these rules, the reader has two indications in every case, to guide him in determining whether the book he has in his hand be the one I describe or not.

This is no unnecessary excess of description. For so frequently, in the sixteenth and seventeenth centuries, were there issues of the same impression under different titles, that all I have done will in many cases only give a presumption as to whether a book in hand is or is not the edition I have described. Were I to begin this work again, I would in every instance make a reference to some battered letter, or defect of lineation, or something which would be pretty certain not to recur in any real reprint. Ordinary errata would not be conclusive: for these might be reprinted for want of perceiving the error.

Rules are given for determining the form of printing by the waterlines of the paper, and by the catchwords. It is supposed that the latter are always at the end of the sheet, and also that the waterlines are perpendicular in folio, octavo, and decimo-octavo books, and horizontal in quarto and duodecimo. But in the first place, a great many old books have catchwords at the bottom of every page, many have none at all; and as to the rule of waterlines, I have found exceptions to every case of it. Pacioli’s Euclid, Venice, fifteen-nine, folio in fours, has horizontal waterlines: the Hypomnemata of Stevinus, an undoubted folio, has thick waterlines both ways.

6. In the smaller type are entered such remarks as suggested themselves on the manner or matter of the work, or on any point arising out of it. In some cases these have extended themselves to short dissertations, such as, on the geometrical foot, page 5,—on Sacrobosco’s knowledge of the Arabic numerals, page 14,—on the invention of + and -, page 19,—on the age of Diophantus, page 47,—on the genuineness of Cocker’s Arithmetic, page 56. But for the most part I have contrived to keep within a very moderate compass as to what is said under each work. [Pg xiv]

The principal point on which I have distinguished one work from another, is as to the use of the old or new method of performing division, which, more than any other single point, decides the character of a work. The letters O, N, or ON, will tell whether the work uses the old, the new, or both. Not that the verbal distinction is here very correct; for neither method is older than the other; and both appear in Pacioli. A description of the now disused method is given by Dr. Peacock, p. 433.

With regard to the works themselves, I have made no selection, as before noticed. No book that I have seen during the compilation has been held too bad to appear; no book that I have not seen too good to be left out. I have had but one discretion to exercise, namely, to determine the extent to which algebra should be considered as arithmetic. In the earlier day, the distinction was slight: I doubt whether I have not overrated it; but it is not an easy line to draw.

The history of Arithmetic, as the simple art of computation, has found little notice from the historians of mathematics in general. They shew themselves deficient in the knowledge of its progress, and of the connexion of that progress with the rest of their subject. The writers whom I can name as having attempted—some more and some less—to supply this defect, are Wallis, Dechâles, Heilbronner, Scheibel, Kästner, Leslie, Delambre, Peacock, and Libri. I speak of the progress of arithmetical writings as works on science, independently of bibliography properly so called, and biography.

Wallis (p. 44 and Additions) was one of those writers whose works remain the standard of the erudition of their day. His algebra, so called, is rather the history, theory, and practice of both arithmetic and algebra. Its miscellaneous and badly indexed form prevents any from knowing what is in it, except those who make a study of it, which none of our day will do, unless they intend to go rather deeply into the history of the exact sciences. But many and many a page by which the writer intended to gain the credit of research, will be found to be a transcript from Wallis. As a connected history, however, it is nothing; and as a bibliography, less. For example, that Regiomontanus used decimal fractions—a very common story—is the consequence of Wallis’s confused method of stating that he introduced the decimal instead of the sexagesimal radius into trigonometry; the confusion arising from his not having a clear knowledge of what had been published of Regiomontanus. And again, we find him, though the editor of an edition of Oughtred, balancing as to [Pg xv] which was written first, Oughtred’s Clavis, or Harriot’s Praxis (which were published in the same year), apparently ignorant that the great method as to which the two authors were chiefly to be compared, did not appear in Oughtred’s first edition at all. But for all this, if Wallis be cautiously watched as to books and dates, his works are most valuable magazines of historical suggestion. From them might be collected a much better scientific history of arithmetic than existed in his time, or, indeed, in any time preceding the publication of Dr. Peacock’s article on the subject.

Claude Francis Milliet Dechâles was a Jesuit, who published, in four very large folios, a complete course of mathematics, including architecture, carpentery, fireworks, and all that was then held to belong to the exact sciences (see page 53). The first volume opens with about a hundred pages (large folio, double column) de progressu matheseos, consisting entirely of description of books, in order of date. The part relating to arithmetic fills nine of these pages. The whole is done with much care, and is, for the mode of describing books current at the time, very accurate; and the opinions given on the books shew that Dechâles had read them. But he is strongly addicted to the very common mistake of judging the books according to what ought to have been said of them, if they had been published in his own time. For example, he finds that there is not sufficient demonstration in Tonstall; which is true, absolutely speaking; but Tonstall is a very Euclid by the side of his contemporaries.

Heilbronner’s Historia, &c. (page 69), though it professes only to give writers up to the beginning of the sixteenth century, makes a particular exception in favour of arithmetic. Up to the year 1740, about 170 authors are recorded, a great many of whom he had not seen. There is also historical dissertation on points of arithmetic. This is a work of great value to the inquirer: he must not rest upon its statements; but he will find more than usual materials for further research.

Scheibel (Additions) may be considered as partly repetition, partly extension, of Heilbronner. He is one of those bibliographers who collect from various sources the names and dates of more editions than those who know catalogues will readily believe in.

Kästner (Additions) falls under the following censure from Dr. Peacock: “The meagre sketch which Kästner has given of some insulated works on the subject, generally contrives to omit almost every [Pg xvi] particular which is essentially connected with the history of the progress of the science.” This is well merited, inasmuch as the author chose to call his work a history, instead of a bibliography; and as the former, nothing can be more incomplete. I have almost as good a right to call my work a history, as Kästner his. But, as a bibliography, it may be urged in defence, that he gave fuller descriptions of books than his predecessors. Scheibel, Heilbronner, and Dechâles sink into title-writers (and bad ones) before Kästner.

The late Professor Leslie (page 89) was one of those men, the strength and asperity of whose opinions would make it fair to deal with them as they dealt with others. In his Philosophy of Arithmetic he has entered incidentally into much of its history. He was, by taste, a searcher of old books; and various dates, &c. occur, which shew that he had more knowledge of books than can be got from catalogues. A few words will sometimes shew this to a person who has compared the books with the accounts of them. But, writing in a popular manner, he does not give references to his authorities, which is a serious diminution of the value of his work. Of Leslie, as an historian on controverted points, one principal thing to be cautious of is, his almost monomaniac antipathy to every thing Hindoo—a most unfortunate turn for an arithmetical investigator. Those who inquire into this subject will see what he is in Colebrooke’s hands; those who do not, may compare his description of the Lilivati, “a very poor performance, containing merely a few scanty precepts,” with the summary of the contents of that work in the Penny Cyclopædia, article Viga Ganita. Leslie also generalises most fearfully every now and then. He informs us that it was the practice throughout Europe to reduce the rules of arithmetic to memorial verses, and that Buckley’s Arithmetica Memorativa appears at one period to have gained possession of the schools and colleges of England. Now the truth is, that the verses attributed to Sacrobosco had never even been printed when Leslie wrote; and Buckley, so far as is known, was printed only once alone, and two or three times as an appendix to a work on logic. Dr. Peacock expresses the truth in saying that, before the invention of printing, the practice of writing memorial verses was common, as appears by manuscript libraries. It is needless to say that, had the practice of using them been common, the presses of the fifteenth and sixteenth centuries would have given them forth in great numbers. But I cannot learn that any metrical work was printed in the fifteenth century, except the Compotus of Anianus, and that only once. [Pg xvii]

Delambre (page 84) wrote on the history of Greek Arithmetic in such a manner as to set that part of the subject fairly going. I doubt if any one since the time of Wallis, at least of their order of note, has exhibited so much of the union of the scholar and the computer. Of the obligations under which the history of astronomy lies to him, it is unnecessary to speak here, or any where: no one man was ever so closely connected with that science, with its past, its present, and its future, by history, observations, and tables.

Of my opinion of Dr. Peacock’s work (page 91) I need hardly say any more, after the pains which I have taken to give reasons on every point in which I differ from its author, and to correct every little error which I have found. This I judge to be the most undoubted compliment which can be paid to any work. Having looked carefully over it, with a great many of the works mentioned in it in my hands, it would be a strong evidence in its favour, were it needed, that I have found no more to set right than is there noted, in matters of dates and circumstances. It is much to be wished that this treatise should be published separately:⁠[4] those who can obtain it will find that it gives life and spirit to the catalogue of books which forms the main part of this work. Up to this moment it is the only work which can be called a history of arithmetic.

M. Libri’s history of science in Italy (page 95) is the work of a man who, to the character of a mathematician, adds that of a man who is well versed in literature, and a successful collector of the rarest works in that of his own country. There is much which is interesting in its early history of Italian Arithmetic. Unfortunately, it is not yet finished; and, I am told, is not likely to be speedily resumed.

The history of most of the sciences resembles a river which sinks underground at a certain part of its course, and emerges again at a distant spot, swelled by certain tributaries, which have joined it in [Pg xviii] the tunnel. Of arithmetic in particular, we note the disappearance in the seventh century of all that was good in the Greek system; and we see the rise, at the invention of printing, of the most trivial part of it, combined with the additions which are now well known to have been received from Eastern sources.

The manuscript literature of the middle ages will prove no very productive source of information, to judge by all that has been made of it hitherto. But then it is to be remembered how seldom, if ever, it has happened, that the investigator of it has united the character of a sufficient mathematician with that of an industrious and well-trained palæographist. Neither the one nor the other can proceed alone: the former has not only to learn how to read, but what to read; for the actual habitat of the manuscripts, and how to get preliminary knowledge of their existence, is a study of itself. The latter is apt to know no difference between what is sound, and what is worthless.

When Euclid, Ptolemy, &c. are first seen to reappear, they come in as Arabic writers. They may be called Greek, but the first translations are from the Arabic; and their effect upon the literature of Europe is, in the first instance, just what it might have been if the authors had been Persians or Saracens. How the communication took place has been considered as a point of small curiosity compared with the importation into Europe of the Arabic numerals.

This subject, both generally, and with reference to the several countries, has been long on the anvil: not the authorship of the letters of Junius has tasked research and ingenuity more than the introduction of the nine digits and the cipher. I suppose nobody would listen to the hypothesis that the former wrote themselves: but I am much inclined to suspect that the latter introduced themselves. The endosmose constantly going on between nations connected by war and commerce would not merely explain so easy a matter, but would render it very difficult to explain how it did not happen, if it had not happened. That the Venetian merchants should not know the system of accounts of those with whom they traded, is incredible: that, at the beginning of the thirteenth century, many priests and some soldiers, returned from their crusades, should not bring back with them the account of so very elementary a difference between themselves and those with whom they had treated, and whom they had in many instances served as slaves taken in war, is unlikely. That Leonard of Pisa, as he is [Pg xix] called, was the first who wrote on or in the new system, is pretty generally affirmed by the Italians of the fifteenth and sixteenth centuries: and there can be no doubt of it. But because we trace the first formal user or expounder to Italy, it by no means follows that the other parts of Europe owe the system to that country in the first instance, however certain it may be that they owe much assistance in the course of the general establishment. It was so common in England in the thirteenth century, that Roger Bacon recommends it (page 14) as a study, of course as a practicable study, for the clergy. Had it not been commonly known, at least as to what it was, he would have done more than give the name of the science, and the names of its rules; he would have added some description. Those who have searched into the matter, merely with a view to Arabic numerals, and without any collateral thoughts about arithmetic in their heads, may have passed over much valuable evidence. They did not know, perhaps, that the organised rules of computation always went with the Arabic system, and never with the Roman or Boethian: that if ever they came to the connected mention of addition, subtraction, multiplication, and division, it ought to have been a sign that they were reading on algorism, as distinguished from arithmetic. This passage of Roger Bacon has been neglected; though the mere occurrence of the word algorism, in a most interpolable clause of one sentence, occurring in Matthew Paris, has received notice and discussion from Mr. Hallam, from a learned writer in the Archæologia, and probably from others.

The rejection of the work attributed to Sacrobosco may be accounted for from the general ignorance of its having been printed under his name at an early period. Though this does not prove the genuineness of the work, it very much adds to the evidence for it. As early as 1523, the learned world was invited to dissent from the assertion that the treatise in question was written by Sacrobosco, and did not do it. It was often cited afterwards, and never, as far as I have seen, with any doubt. But the only writer of this or the last century that I can find, who describes the printed edition, is the Italian editor of Fabricius, Bibliotheca Latina.

Come how they might, however, we find ourselves, at the invention of printing, in possession of the knowledge that two distinct systems of arithmetic were current. The streams which had united in one bed had not mingled their waters; and for nearly a century two distinct classes of writings present themselves. Their only common point is the use of the Arabic or Indian numerals and method of notation. [Pg xx]

The first, or algoristic system, as we may call it, proceeded by systematic rules to the performance of questions of computation. It embraced mercantile arithmetic, and what was then called algebra: and laborious and prolix efforts were made to combine the two; that is, to represent in such a form as we should now call algebraic, as distinguished from arithmetical, the solutions of questions of commerce, or of what might become such, if the connexion were diligently fostered. I cannot suppose that the multifarious problems of exchange of different kinds which abound in Pacioli and his immediate followers, were actually useful to the merchant: but there is an obvious leaning to the idea that they ought to be so. Geometry was also an application of this arithmetic; and in a manner which strongly marks its eastern character, as will appear to those who compare the work of Pacioli with the Indian books.

The second, or Boethian system, as I shall call it, because the work of Boethius was its great text-book, did not give any rules of calculation, nor apply itself to any application. The study was the properties of numbers, and particularly of their ratios. There was no art about it; and we have no means of telling whether the philosophers of this school reckoned on their fingers, or used an abacus, or put pen to paper for the performance of some organised method of computation. To judge by the smallness of the numbers used in the instances adduced, we must suppose that the writers left it to their readers to do as they liked best.

For some specimens of the laborious manner by which the Pythagorean Greeks, in the first instance, and afterwards Boethius in Latin, had endeavoured to systematise the expression of numerical ratios, I may refer the reader to the article “Numbers, old appellations of,” in the Supplement to the Penny Cyclopædia. If I were to give any account of the whole system, on a scale commensurate with the magnitude of the works written on it, the reader’s patience would not be subquatuordecupla subsuperbipartiens septimas—or, as we should now say, seven per cent—of what he would find wanted for the occasion. Not that the books I am speaking of get quite so far as this. I am hardly prepared to say exactly what number under fifty ought to be named as the terminus of the Boethian store of numbers; but certainly we rarely find them choose their instances from numbers above it. The system broke down under its own phraseology, as did that of the Yancos [Pg xxi] mentioned by Dr. Peacock, who could not get further than three, because they could not express this idea by any thing more simple than Poettarrarorincoaroac.

The Italian school of algorists, with Pacioli at their head, found followers in Germany, England, France, and Spain; and in all but England, the Boethian school also. I cannot discover a single English work which pays any detailed attention to the latter class of arithmeticians. Tonstall, and still more Recorde, for the former appears to have been little known, were the preservers; and before the end of the sixteenth century the ordinary style of commercial arithmetic, which has prevailed among us ever since, was in course of establishment.

This gradual formation of the English school of commercial works will be apparent enough in my list. From the time of Recorde, we always were conspicuous in numerical skill as applied to money. The questions of the English books are harder, involve more figures in the data, and are more skilfully solved. It is possible that I might, if my French, German, and Italian list were more complete, produce exceptions to this rule. But nothing, I think, could arise to alter my conviction that the efforts which were made in this country towards the completion of the logarithmic tables in the seventeenth century, and the instantaneous appreciation of the value of the discovery of logarithms, were the result of that superiority in calculation which I assert to have been formed in the sixteenth. And yet this last-named century produced one man in France, one in Germany, and one in Italy, with either of whom no one English calculator could compare in extent of operations: I allude to Vieta, Rheticus, and Cataldi. There was no opportunity to compete with these men; for the subjects on which they worked were not introduced here. It was only towards the end of the sixteenth century that what were then the higher parts of the mathematical sciences began to be disseminated with effect in Britain.

To the commercial school of arithmeticians above noted we owe the destruction of demonstrative arithmetic in this country, or rather the prevention of its growth. It never was much the habit of arithmeticians to prove their rules: and the very word proof, in that science, never came to mean more⁠[5] than a test of the correctness of a particular operation, by reversing [Pg xxii] the process, casting out the nines, or the like. As soon as attention was fairly diverted to arithmetic for commercial purposes alone, such rational explanation as had been handed down from the writers of the sixteenth century began to disappear, and was finally extinct in the work of Cocker, or Hawkins, as I think I have shewn reason for supposing it should be called. From this time began the finished school of teachers, whose pupils ask, when a question is given, what rule it is in, and run away, when they grow up, from any numerical statement, with the declaration that any thing may be proved by figures—as it may, to them. Any thing may be unanswerably propounded, by means of figures, to those who cannot think upon number. Towards the end of the last century, we see a succession of works, arising one after the other, all complaining of the state into which arithmetic had fallen, all professing to give rational explanation, and hardly one making a single step in advance of its predecessors.

It may very well be doubted whether the earlier arithmeticians could have given general demonstrations of their processes. It is an unquestionable fact of observation, that the application of elementary principles to their apparently most natural deductions, without drawing upon subsequent, or what ought to be subsequent, combinations, seldom takes place at the commencement of any branch of science. It is the work of advanced thought. But the earlier arithmeticians and algebraists had another difficulty to contend with: their fear of their own half-understood conclusions, and the caution with which it obliged them to proceed in extending their half-formed language. It was not merely by oversight, I suspect, that Oughtred so often calculates ab + ac by two multiplications, instead of using a(b + c): but rather from that same general fear of abbreviation, and suspicion that error may lurk in it, which possesses men of business who dare not multiply by 10 by the annexation of a cipher, but proceed with each figure, and carriage, as they would do if the multiplier were 8 or 7. I have seen this often enough; and things nearly as strange times without number. But what shall we say to the following; a most sufficient recommendation of the study of old works to the teacher, as shewing that the difficulties which it is now (I speak to the teacher not the rule-driller) his business to make smooth to the youngest learners, are precisely those which formerly stood in the way of the greatest minds, and sometimes effectually stopped their progress. Perhaps no man of his day had so much power over mathematical language as Wallis. But the following [Pg xxiii] extract of a letter from him to Collins (Macclesfield Collection, vol. ii. p. 579), in 1673, shews that he once had doubts whether he might dare write down the square root of 12 as being twice the square root of 3, however certain he might be that it is so; because no one had so written it. Speaking of the square root of a negative quantity, he says, “Only, though I had from the first a good mind to it, I durst not without a precedent, when I was so young an algebraist as in the history my late letter reports, take upon me to introduce a new way of notation, which I did not know of any to have used before me. And it was not without some diffidence that I ventured on 2√3 instead of √12 prefixed to a surd root; but I found it so expedient, not only for the discovering the root of a binomial, whether quadratic, cubic, or others, but for the adding and subducting of commensurable surds, that I resolved to use it for my own occasions, before I knew whether others would approve of it or no; especially having found in the first edition of Oughtred’s Clavis (for I had not then seen the second) one instance or two for it, to justify myself if it should be questioned. But since that time it is grown more common, and I perhaps have somewhat contributed thereunto.”

If I were to take such things from old and unknown writers, in which they abound, I should be thought to waste the time of my readers by giving an undue importance to the difficulties of very inferior minds. But those who do not want such a confession as the above from a Wallis, to help to persuade them that the difficulties of progress are of the same character in all ages, will find in old works plentiful and useful supplies of thought for a modern teacher.

Another instance of the halting progress of language is the mode of introduction of the decimal point. Stevinus, the undoubted introducer of the decimal fraction (though others may have seen very clearly the great use of 10, 100, 1000, &c. as divisors), had no such thing, but only a distinct mode of denoting the meaning of each decimal place. Dr. Peacock mentions Napier as being the person to whom the introduction is unquestionably due: a position which I must dispute, upon additional evidence.

The inventor of the single decimal distinction, be it point or line, as in 123·456 or 123|456, is the person who first made this distinction a permanent language: not using it merely as a rest in a process, to be useful in pointing out afterwards how another process is to come [Pg xxiv] on, or language is to be applied; but making it his final and permanent indication as well of the way of pointing out where the integers end and the fractions begin, as of the manner in which that distinction modifies operations. Now, first, I submit that Napier did not do this; secondly, that if he did do it, Richard Witt did it before him (page 34).

I have not seen Wright’s summary of logarithms, of 1616, to which Dr. Peacock refers for some indications of the decimal point. But I take with confidence his assertion that the first distinct notice of the thing in question, by or by means of Napier, is contained in the Rabdologia (page 35).

In this famous tract there are but two instances to the purpose: a scanty number, be it noticed, for a person who had seized the idea of completing the decimal scale. The first is an instance of division which Dr. Peacock cites. Here the quotient is, as we should now say, 1993·273. Napier uses a comma in his quotient, as a rest, and writes 1993,273, and then presents his answer in the form 19932′7″3‴, as Stevinus would have done. Dr. Peacock states this, and notes it as a “partial conformity to the practice of Stevinus.” Unfortunately Napier has not given us instances enough to tell us whether he had a practice of his own: and in the only other case, which Dr. Peacock also cites (I have verified both citations), he adheres still more closely to the method of Stevinus. For though, in a question of addition, he has drawn a line down the interval where our decimal points would be, which line would be as distinctive in his total as in his addenda, he places the exponents of Stevinus in this total, which thus has both distinctions, and stands 1994|9′1″6‴0⁗.

I cannot trace the decimal point in this: but if required to do so, I can see it more distinctly in Witt (page 33), who published four years before Napier. But I can hardly admit him to have arrived at the notation of the decimal point. For though his tables are most distinctly stated to contain only numerators, the denominator of which is always unity followed by ciphers; and though he has arrived at a complete and permanent command of the decimal separator (which with him is a vertical line) in every operation, as is proved by many scores of instances; and though he never thinks of multiplying or dividing by a power of 10 in any other way than by altering the place of this decimal separator; yet I cannot see any reason to suppose that he gave a meaning to the quantity with its separator inserted. I apprehend that if asked what his 123|456 was, he would have answered: [Pg xxv] It gives 123-456/1000, not it is 123-456/1000. This is a wire-drawn distinction: but what mathematician is there who does not know the great difference which so slight a change of idea has often led to? The person who first distinctly saw that the answer -7 always implies that the problem requires 7 things of the kind diametrically opposed to those which were assumed in the reasoning, made a great step in algebra. But some other stepped over his head, who first proposed to let -7 stand for 7 such diametrically opposite things.

Who, then, was the real inventor of the decimal separator, to give both interpretation and operative form? This is a question I will not answer positively: nor attempt to answer till I have pointed out how the case stands with several possible claimants. And first, as to Briggs. He does not indeed arrive at the simple decimal point (which is a strong presumption against such a thing being suggested by Napier; for who would have learnt it from him, if not Briggs?)—but he omits the denominator, and draws a line under the decimals. And it is a further presumption against any such idea of abbreviation being common, that Briggs explains his notation as quite peculiar to himself. In the preface, or Ad Lectorem, of the Arithmetica Logarithmica, London, sixteen-twenty-four, folio in twos, he commences thus:

“Ut obscuritatis crimen, quantum in me situm fuerit effugiam, paucula hæc humanissime Lector, te admonendum censui. 1. Si numerus occurrat cui linea subscribitur, notæ illæ quæ supra lineam sunt descriptæ, Numeratorem partium constituunt, quarum Denominator semper intelligitur unitas, cum tot cyphris, quot sunt notæ superius positæ. ut 75 designant ⁷⁵/₁₀₀ vel ¾. sic 5 9321 designant

Oughtred adopted both the vertical and sub-horizontal separators, thus shutting up the numerator in a semi-rectangular outline. In the mean time, Gunter, after adopting Briggs’s notation in the first instance, gradually dropt it, and substituted the decimal point. The history of this symbol in his hands is rather curious, on account of its having been so completely overlooked, though all the circumstances were to be seen in widely circulated works.

Gunter published in 1620 his Canon Logarithmorum, sines and tangents on Briggs’s system. To this table there is no explanation. But an explanation was written by him, and, though I cannot make out that it was ever published separately, yet it appears in the collection of his works. I have what is called the second edition of this collection, [Pg xxvi] London, sixteen-thirty-six, quarto, which, for various reasons, I suspect to be the first, thinking that the announcement of second edition merely refers to the fact of its having been previously published during Gunter’s life. After the finis, with new paging and signatures, comes ‘The Generall use of the Canon and Table of Logarithmes.’ This I suspect to have then made its first appearance: but from the very manner in which the decimal fractions are treated, it is clear that it was written before or with the work on the sector, &c., which first⁠[6] came out in sixteen-twenty-three; and this, though it makes reference to that work, or at least to its matter. At the beginning of this ‘Generall use, &c.’ common fractions are used, even when the denominators are decimal. At page 13, Briggs’s notation appears, without explanation: and 116 04 is the third proportional to 100 and 108; this continues through page 14. In page 15, a dot is added to Briggs’s notation in one instance: 100l. in 20 years at 8 per cent becomes 466. 095l. At the bottom of this page Briggs’s notation disappears thus, “It appeareth before, that 100l. due at the yeares end is worth but 92 592 in ready money: If it be due at the end of 2 yeares, the present worth is 85l. 733: then adding these two together, wee have 178l. 326 for the present worth of 100. pound annuity for 2 yeeres and so forward.” After this change, thus made without warning⁠[7] in the middle of a sentence, Briggs’ notation occurs no more in the part which relates to numbers. But in the following chapter, which is trigonometrical (and which may have been written first, for Gunter’s own logarithms are only trigonometrical), it reappears, sometimes with and sometimes without the dot. In the previous work on the sector, &c., the simple point is always used: but in explanation the fraction is not thus written, but described as parts. Thus, 32.81 feet in operation is 32 feet 81 parts in the description of the question or answer.

It was long before the simple decimal point was fully recognised in all its uses, in England at least: and on the continent the writers were rather behind ours in this matter. As long as Oughtred was widely used, that is, till the end of the seventeenth century, there must always have been a large school of those who were trained to the notation [Pg xxvii] 123|456. To the first quarter of the eighteenth century, then, we must refer, not only the complete and final victory of the decimal point, but also that of the now universal method of performing the operation of division and extraction of the square root.

This evident slowness in the admission of such important improvements will not appear singular to those who observe what is now taking place with reference to Horner’s method for the solution of equations. It will be the business of some successor of mine, a century or two hence, to distinguish in his list, by some appropriate mark, the works of our period which adopt this method from those which do not: just as I have done with the two methods of division. I hope, if I live to publish a second edition, that I may be able to add to my list the name of the first work, written for students in the University of Cambridge, which shall contain this most fundamental addition to the processes of pure arithmetic: which, though it has found its way into the public examinations, is not yet in the books which prepare students for them.

This has been a long digression, on what is perhaps the most interesting part of the subject, its language. We owe every thing, almost, to the simplicity of certain modes of expression. Nothing is more clear than that the Greek geometers, with all the acuteness and perseverance necessary to carry results of arithmetical application to a high pitch, and all the taste for numbers which would turn their thoughts that way, were stopped by their insufficient system of numeration, and the tediousness of their processes. The history of language, then, is of the highest order of interest, as well as utility: its suggestions are the best lessons for the future which a reflecting mind can have.

In looking over the list of books, it will be observed, that during the eighteenth and nineteenth centuries there is a great preponderance of English writers. The law I imposed on myself, of entering no books which I could not see, necessarily brought this about. During the last two centuries, the elementary works of the different countries in Europe have not gained great general circulation. They have ceased to be written by the greatest names, for the most part; and they have rarely been cited by the historians of science. Copies which have found their way to this country have probably soon been destroyed. As it is my intention to endeavour to extend this list, whether I publish the extensions or not, and in any case to provide for the preservation of the materials I collect, it will be worth the while of any one who is [Pg xxviii] able, to furnish me with information on works which I have not seen. I think it probable that any one who has had the curiosity to rescue three books of arithmetic from a stall, will find that one of them is not in this list. From any such person I shall thankfully receive the full title of the neglected work, with the form in which it is printed. Nor need it by any means be presumed that because a book is wholly unknown, it proves nothing in the history of the science. A book so thoroughly lost as that of Witt, contains a nearer approach to the decimal point than was made by Napier. Horner found a close approximation to his own method, for the case of the cube root, in an obscure compendium of arithmetic. I might also instance Dary’s discovery mentioned in page 48; and other things of the same kind. The history of hints given before the time at which they were (perhaps could be) made to bear fruit, would be a very curious one: and the progress of science will never be well understood until some little account can, in each case, be given of the reason why a notion should be so productive at a particular period, which was so barren at a previous one.

Florence, fourteen-ninety-one. Philip Calandri. ‘Philippi Calandri ad nobilem et studiosum Julianum Laurentii Medicem de Arimethrica opusculum.’ Octavo (small).

[Pg 2]

Venice, M.cccc.lxliiii., fourteen-ninety-four. Lucas Pacioli, de Burgo Sancti Sepulcri. ‘Summa de Arithmetica Geometria Proportioni et Proportionalita.’ Folio in fours.

Tusculano, fifteen-twenty-three. Lucas Pacioli. ‘Summa de Arithmetica geometria. Proportioni: et proportionalita: Novamente impressa In Toscolano su la riva dil Benacense et unico carpionista Laco: Amenissimo Sito: de li antique et evidenti ruine di la nobil cita Benaco ditta illustrato: Cum numerosita de Imperatorii epithaphii di antique et perfette littere sculpiti dotato: et continens⁠[8] finissimi et mirabil colone marmorei: inumeri fragmenti di alabastro porphidi et serpentini. Cose certo lettor mio diletto oculata fide miratu digne sotterra se ritrovano.’ Folio in fours (ON).

No place, no date. John Muris. ‘Arithmetices Compendium ex Boetii Libris per Johannem Muris excellentis ingenii virum Accurate congestum.’ Quarto.

Venice, fifteen-one. Geo. Valla. ‘De expetendis et fugiendis rebus opus.’ Folio in fours.

Cologne, fifteen-one. John Huswirt. ‘Enchiridion Novus Algorismi summopere visus De integris. Minutiis vulgaribus Projectilibus Et regulis mercatorum sine figurarum (more ytalorum) deletione percommode tractans,’ &c. Quarto, twos and threes (O).

Paris, fifteen-three. Joh. Faber Stapulensis, Jodocus Chlichtoveus, and Carolus Bovillus. ‘In hoc libro contenta [Pg 4] Epitome compendiosaque introductio in libros Arithmeticos divi Severini Boethii,’ &c. &c. Folio in fours.

Basle, fifteen-eight. Gregorius Reisch. ‘Margarita Philosophica, cum additionibus novis: ab auctore suo studiosissima revisione tertio superadditis.’ Quarto in fours.

• 1  breadth of a barleycorn.
• 4 =  1 digit.
•   16 =  4 =   1 palm (across the middle of the fingers).
•   64 =   16 =   4 = 1 foot.
•   96 =   24 =   6 = 1½ = 1 cubit.
• 160 =   40 = 10 = 2½ = 1⅔ = 1 step (gressus).
• 320 =   80 = 20 =   5 =  3⅓ = 2 = 1 pace.
• 640 = 160 = 40 = 10 =  6⅔ = 4 = 2 = 1 perch (pertica).
•  
• 125 paces 1 Italic stadium.
• 8 stadia 1 Italic mile.
• 4 Italic miles a German mile.
• 5 Italic miles a Swiss mile.

Margarita Philosophica, above described. In the Strasburg edition of fifteen-four, the length of the geometrical palm is less than 2½ inches by from half to three fourths of the 24th of an inch. Taking it half way between these, the four-palm foot is 9·9 inches English. In the Basle edition of fifteen-eight, in which the woodcuts are of much rougher execution, the palm is 2·64 inches, giving a foot of 10·56 inches. The palm only is given in both cases.

Oppenheim, fifteen-twenty-four. Stöffler. ‘Elucidatio Fabricæ Ususque Astrolabii.’ Folio in threes (quarto size).

The digit, palm, and foot, are separately given; the foot is divided into palms, and all agree excellently well with one another. The foot is exactly 9·75 inches English.

Paris, fifteen-twenty-six. John Fernel. ‘Monalosphærium.’ Folio in threes.

Paris, fifteen-twenty-eight. John Fernel. ‘Cosmotheoria.’ Folio in threes.

Paris, fifteen-fifty-two. Jac. Koebelius. ‘Astrolabii Declaratio.’ Octavo.

Frankfort, sixteen-twenty-one. Peter Ryff. ‘Questiones Geometricae.’ Quarto.

[Pg 10]

Paris, fifteen-fourteen. Boethius, Jordanus Nemorarius, Shirewode (?), J. Faber (Stapulensis). ‘In hoc opere contenta Arithmetica ... Musica ... epitome ... Boethii: rithmimachie ludus qui et pugna numerorum appellatur.’ Folio in fours. Second edition.

Paris, fifteen-fourteen. Nicolas Cusa. ‘Hæc accurata recognitio trium voluminum operum Clariss. P. Nicolai Cusæ Card. &c.’ In three volumes, folio in fours.

Vienna, fifteen-fourteen. Joh. Cusanus. ‘Algorithmus linearis projectilium, de integris perpulchris arithmetrice artis regulis,’ &c. Quarto size.

A tract of seven pages on counters.

Augsburg, fifteen-fourteen. Jacob Kobel (printer). ‘Ain Nerv geordnet Rechen biechlin auf den linien mit Rechen pfeningen:’ &c. Quarto, twos and threes (duodecimo size).

[Pg 11] Oppenheym, fifteen-fifteen. Jacob Kobel. ‘Eyn New geordnet Vysirbüch.’ Quarto in threes.

A work on gauging, in which the Arabic numerals are used.

Vienna, fifteen-fifteen. Boethius, J. de Muris, Thos. Bradwardin, Nic. Horem, Peurbach, de Gmunden. ‘Contenta in hoc libello. Arithmetica communis, Proportiones breves, de latitudinibus formarum, Algorithmus D. Georgii Peurbachii in integris, Algorithmus Magistri Joh. de Gmunden de minuciis physicis.’ Quarto.

Paris, fifteen-fifteen. Gaspar Lax. ‘Arithmetica speculativa magistri Gasparis Lax Aragonensis de sarinyena duodecim libris demonstrata.’ Also, same date and place, ‘Proportiones magistri Gasparis,’ &c. Folio in threes.

Lyons, fifteen-fifteen. John de Lortse (Dominican). ‘Oeuvre tressubtile et profitable de lart et science de aristmeticque: et geometrie translate nouvellement despaignol en francoys Auquel est demonstre par figure evidemment: tant le nombre entier: nombre rompu: regie de compaignies: soubde fin: que toutes aultres choses qui par geometrie et aristmeticque peuvent estre comprises: comme appert par la table cy apres mise.’ Quarto (O) (166 folios).

Venice, fifteen-twenty. R. Suiseth. ‘Calculator. Subtilissimi Ricardi Suiseth Anglici Calculationes noviter emendate atque revise.’ Folio in threes.

Lyons, fifteen-twenty. Stephen de la Roche Villefranche. ‘Larismethique nouvellement composee divisee en deux parties dont la premiere tracte des proprietes parfections et regles de la dicte Science,’ &c. Folio in fours (O).

Paris, fifteen-twenty-one. Boethius. ‘Divi Severini Boetii Arithmetica duobus discreta libris: adjecto commentario, mysticam numerorum applicationem perstringente, declarata.’ Folio in threes.

London, fifteen-twenty-two. Tonstall (Bp. of London). ‘De Arte supputandi libri quatuor Cutheberti Tonstalli.’ Quarto. (Printed by Pynson.)

Paris, fifteen-twenty-nine. The same title. Quarto in fours. (Printed by Rob. Stephens.)

Strasburg, fifteen-forty-four. ‘De Arte ... Tonstalli, hactenus in Germania nusquam ita impressi.’ Octavo.

Venice, fifteen-twenty-three. Sacrobosco. ‘Algorismus domini Joh. de Sacro Busco noviter impressum.’ Quarto.

Paris, fifteen-twenty-six. John Martin Siliceus. ‘Arithmetica Joh. Martini Silicei theoricen praxinque luculenter complexa’ &c. Folio in threes.

Paris, fifteen-twenty-eight. John Fernel. ‘De proportionibus Libri duo.’ Folio in threes.

[Pg 16]

Leyden, fifteen-thirty-one. Joach. Fortius Ringelbergius. ‘De Ratione Studii.’ Octavo (duodecimo size).

Venice, ——. Giovanni Sfortunati. ‘Nuovo Lume, Libro di Arithmetica ... Composto per lo acutissimo prescrutatore delle Archimediane, et Euclidiane dottrine Giovanni Sfortunati da Siena.’ Quarto in fours.

Nuremberg, fifteen-thirty-four. ‘Algorithmus demonstratus.’ Quarto.

Paris, fifteen-thirty-five. Orontius Fineus. ‘Arithmetica practica libris quatuor absoluta ... Recens ab Authore castigata ... hoc est in nativum splendorem (quem priorum impressorum amiserat incuria) summa fidelitate restituta.’ Folio in fours.

Strasburg, fifteen-thirty-six. Hudalrich or Huldrich Regius. ‘Utriusque Arithmetices Epitome ex variis authoribus concinnata per Hudalrichum Regium.’ Octavo (duodecimo size).

Paris, fifteen-thirty-eight. Nicomachus of Gerasa. ‘Νικομαχου Γερασινου Ἀριθμητικης βιβλια δυο Nicomachi Gerasini Arithmeticæ Libri duo. Nunc primum typis excusi, in lucem eduntur. Parisiis in ofiicina Christiani Wecheli.’ Quarto.

Milan, fifteen-thirty-nine. Jerome Cardan. ‘Practica Arithmetice et mensurandi singularis.’ Octavo (O).

London. ‘This boke sheweth the maner of measurynge of all maner of lande, as well of woodlande, as of lande in the felde, and comptynge the true nombre of acres of the same. [Pg 18]

Newlye invented and compyled by Syr Rycharde Benese, Chanon of Marton Abbay besyde London. Prynted in Southwarke in Saynt Thomas hospitall, by me James Nicolson.’ Quarto.

Again—London. ‘The Boke of measuryng of Lande as well of Woodland as Plowland, and pasture in the feelde: and to compt the true nombre of Acres of the same. Newly corrected and compiled by Sir Richarde de Benese. ¶ Impynted at London, by Thomas Colwell.’

Paris, fifteen-thirty-nine. Johannes Noviomagus. ‘De Numeris libri duo.’ Octavo (small) (O).

Paris, fifteen-forty-four. ‘Orontius Finæus. Arithmetica Practica ... multisque accessionibus locupletata.’ Octavo (O). [Pg 19]

Paris, fifteen-fifty-five. Orontius Finæus. ‘De Arithmetica Practica libri quatuor.’ Quarto (O).

Nuremberg, fifteen-forty-four. Michael Stifel. ‘Arithmetica integra. Authore Michaele Stifelio. Cum præfatione Philippi Melancthonis.’ Quarto. (Hutton’s Tracts, v. ii. p. 237.)

No place, nor date. Wm. Buckley. ‘Arithmetica memorativa sive compendaria Arithmeticæ tractatio.’ Apparently octavo.

Basle, fifteen-fifty. John Scheubel. ‘Euclidis Megarensis ... Algebræ porro Regulæ ... his libris præmissæ sunt....’ Folio in twos.

Leyden (Lugduni, without further description), fifteen-fifty. Henry Cornelius Agrippa. ‘De Occulta Philosophia Libri Tres.’ Octavo (Italic letter).

Augsburgh, fifteen-fifty-four. Psellus. ‘De quatuor scientiis mathematicis.’ Edited by Xylander. Gr. Lat. Octavo (duodecimo size).

Venice (the first two parts fifteen-fifty-six, the rest fifteen-sixty). Nicolas Tartaglia. ‘La prima parte dei numeri e misure aritmetica geometria et algebra.’ Folio in threes (O).

London, fifteen-fifty-seven. Robert Recorde. ‘The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Cossike practise, with the rule of Equation: and the woorkes of Surde Numbers.’ Quarto.

Lione, fifteen-fifty-eight. Giov. Franc. Peverone. ‘Due [Pg 22] brevi e facili trattati. Il primo d’Arithmetica: l’altro di Geometria,’ &c. Quarto.

Leyden, fifteen-fifty-nine. Joh. Buteo. ‘Joan. Buteonis Logistica quæ et Arithmetica vulgo dicitur.’ Quarto in fours. (O).

Paris, fifteen-sixty. James Pelletier. ‘Jacobi Peletarii Cenomani, de occulta parte numerorum, quam algebram vocant, libri duo.’ Quarto. [First edition, fifty-eight.]

London, fifteen-sixty-one. Robert Recorde. ‘The Grounde of artes: Teaching the worke and practise of Arithmetike, both in whole numbres and Fractions, after a more easyer and exacter sorte than any like hath hitherto been sette forthe: Made by M. Robert Recorde, Doctor of Physik, and now of late ouerseen and augmented with new and necessarie Additions.

Octavo (duodecimo size) (O).

[Pg 23] Vienna, fifteen-sixty-one. Christoffer Rudolff. ‘Kunstliche rechnung mit der ziffer und mit der zalpfenningen,’ &c. Octavo (small).

Antwerp, fifteen-sixty-five. Valentine Menher de Kempten. ‘Practicque pour brievement apprendre à Ciffrer, et tenir livre de Comptes, avec la regle de Coss, et Geometrie.’ Octavo. [Date at the end, sixty-four, July.]

Antwerp, fifteen-sixty-seven. Pedro Nunez. ‘Libro de algebra en arithmetica y geometria.’ Octavo (small).

London, fifteen-sixty-eight. Humfrey Baker. ‘The Well Spryng of Sciences, which teacheth the perfecte woorke and practise of Arithmeticke ... beautified with moste necessary rules and questions....’ Octavo (very small size) (O).

Paris, fifteen-seventy-one. Alex. Vandenbussche. ‘Arithmetique militaire.’ Quarto. [Pg 24]

London, fifteen-seventy-one. Leonard Digges. ‘A Geometricall Practise, named Pantometria ... framed by Leonard Digges, Gentleman, lately finished by Thomas Digges his sonne. Who hath also thereunto adjoyned a Mathematicall Treatise of the five regulare Platonicall bodies,’ &c. Quarto. Reprinted as follows: London, fifteen-ninety-one. ‘A Geometrical practical treatize named Pantometria.... First published by Thomas Digges Esquire ... lately reviewed by the author himself,’ &c. Quarto.

Venice, fifteen-seventy-five. Franc. Maurolycus. ‘Arithmeticorum libri duo.’ Quarto in fours.

Basle, fifteen-seventy-nine. Christian Urstisius. ‘Elementa Arithmeticæ, logicis legibus deducta.’ Octavo (duodecimo size) (O).

Brescia, fifteen-eighty-one. Ant. Maria Vicecomes (Visconti). ‘Practica numerorum et mensurarum,’ &c. Quarto. [Pg 25]

Antwerp, fifteen-eighty-one. Gemma Frisius. ‘Arithmeticæ practicæ methodus facilis, in eandem J. Steinii et J. Peletarii annotationes,’ &c. Octavo (small size) (O).

Paris, fifteen-sixty-one. ‘Arithmeticæ Practicæ Methodus facilis, per Gemmam Frisium ... jam recens ab ipso authore emendata, et multis in locis insigniter aucta. Huc accesserunt Jacobi Peletarii Cenomani annotationes ... Quibus demum ab eodem Peletario additæ sunt Radicis utriusque demonstrationes.’ Octavo (small) (O).

London, fifteen-eighty-three. H. Baker, ‘The Well spring of sciences. Which teacheth the perfect worke and practise of Arithmeticke ... set forthe by Humfrey Baker, Londoner, 1562. And nowe once agayne perused augmented and amended in all the three partes, by the sayde Aucthour.’ Octavo (duodecimo size).

London, fifteen-eighty-four. John Blagrave. ‘The mathematical jewel.’ Folio in twos.

Leyden, fifteen-eighty-five. Simon Stevinus. ‘L’Arithmetique de Simon Stevin de Bruges: Contenant les computations des nombres arithmetiques ou vulgaires: Aussi l’algebre, avec les equations de cinc quantitez. Ensemble les quatre premiers livres d’Algebre de Diophante d’Alexandrie maintenant premierement traduicts in François. Encore un livre particulier de la Pratique d’Arithmetique, contenant entre autres, Les Tables d’Interest, La Disme; Et un traicté des Incommensurables grandeurs: Avec l’Explication du Dixiesme livre d’Euclide.’ Octavo.

Leyden, sixteen-thirty-four. Albert Girard. ‘Les Oeuvres Mathematiques de Simon Stevin de Bruges,’ &c. Folio in threes.

London, fifteen-eighty-eight. John Mellis. ‘A briefe instruction and maner hovv to keepe bookes of Accompts after the order of Debitor and Creditor, and as well for proper Accompts partible, &c. By the three bookes named the Memoriall Journall and Leager, and of other necessaries appertaining to a good and diligent marchant. The which of all other reckoninges is most lawdable: for this treatise well and sufficiently knowen, all other wayes and maners may be the easier and sooner discerned, learned and knowen. Newely augmented and set [Pg 28] forth by John Mellis Scholemaister, 1588. Imprinted at London by John Windet, dwelling at the signe of the white Beare, nigh Baynards Castle. 1588.’ Octavo.

London, fifteen-ninety. Cyprian Lucar. ‘A treatise named Lucarsolace,’ &c. Quarto.

London, fifteen-ninety. Thomas Digges. ‘An Arithmetical warlike treatise named Stratioticos ... first published by Thomas Digges Esquire, anno Salutis, 1579,’ &c. Quarto (O).

Helmstadt, fifteen-ninety. Heizo Buscher. ‘Arithmeticæ libri duo.’ Octavo (small) (O).

Bergomi, fifteen-ninety-one. Peter Bungus. ‘Numerorum Mysteria.’ Second edition. Quarto in fours.

Frankfort, fifteen-ninety-one. Jordano Bruno. ‘De Monade [Pg 29] numero et Figura liber Consequens Quinque de Minimo magno et Mensura. Item de Innumerabilibus, immenso, et Infigurabili; seu De Universo et Mundis libri octo.’ Octavo (duodecimo size).

Frankfort, fifteen-ninety-two. Peter Ramus and Lazarus Schoner. ‘Petri Rami Arithmetices Libri Duo, et Algebræ totidem: a Lazaro Schonero emendati et explicati. Ejusdem Schoneri libri duo: alter, De Numeris figuratis; alter, De Logistica Sexagenaria.’ Octavo (O).

London, fifteen-ninety-two. Thomas Masterson, ‘his first booke of Arithmeticke.’

London, fifteen-ninety-two. Thomas Masterson, ‘his second booke of Arithmeticke.’

London, fifteen-ninety-four. Thomas Masterson, ‘his Addition to his first booke of Arithmetick.’

London, fifteen-ninety-five. Thomas Masterson, ‘his thirde booke of Arithmeticke.’ All Quarto (O).

[Pg 30] London, fifteen-ninety-four. Blundevile. ‘M. Blundevile His Exercises, containing sixe Treatises,’ &c. Quarto in fours.

London, fifteen-ninety-six. ‘The Pathway to Knowledge. Conteyning certaine briefe Tables of English waights, and Measures ... With the Rules of Cossicke, Surd, Binomicall, and Residuall Numbers, and the Rule of Equation, or of Algebere ... And lastly the order of keeping of a Marchants booke, after the Italian manner, by Debitor and Creditor ... Written in Dutch, and translated into English, by W. P.’ Quarto (O).

Alcmaar (preface dated from Amsterdam), fifteen-ninety-six. Nicolaus Petri. ‘Practicque omte Leeren Rekenen Cypheren ende Boeckhouwen met die Reghel Coss....’ Octavo (small) (O).

Venice, fifteen-ninety-nine. Joh. Bapt. Benedictus. ‘Speculationum Liber.’ Folio in twos.

London, sixteen-hundred. Thomas Hylles. ‘The arte of vulgar arithmeticke, both in integers and fractions, devided into two bookes ... Nomodidactus Numerorum ... Portus Proportionum ... whereunto is added a third book entituled Musa Mercatorum, comprehending all rules used in the most necessarie and profitable trade of merchandise ... Newly collected, digested, and in some part devised, by a wel willer to the Mathematicals.’ Quarto (O).

Paris, sixteen-hundred. John Chambers. ‘Barlaami Monachi Logistica nunc primum Latiné reddita et scholiis illustrata.’ Quarto.

Leyden, sixteen-three. C. Dibuadius. ‘In Geometriam Euclidis prioribus sex Elementorum libris comprehensam Demonstratio Numeralis.’ Quarto.

Leyden, sixteen-five. Simon Stevinus. ‘Tomus secundus Mathematicorum hypomnematum de Geometriæ praxi.’ Folio in threes. (The printing delayed beyond the title-date.)

[Pg 33] Augsburgh, sixteen-nine. Geo. Henischius. ‘Arithmetica perfecta et demonstrata.’ Quarto (ON).

Mayence, sixteen-eleven. Christopher Clavius. ‘Epitome Arithmetices Practicæ.’ Folio in threes (O).

Bologna, sixteen-thirteen. Pietro Antonio Cataldi. ‘Trattato del modo brevissimo di trovare la Radice quadra delli numeri.’ Folio.

The rule for the square root is exhibited in the modern form, and Cataldi shews himself a most intrepid calculator. But the greatest novelty of the book is the introduction of continued fractions, then, it seems, for the first time presented to the world. Here, with great labour, but still with success, Cataldi reduces the square roots of even numbers to continued fractions of the form

He then uses these fractions in approximation, but without the assistance of the modern rule by which each approximation is educed from the preceding two. Thus he reduces, among other examples, the square root of 78 to

London, sixteen-thirteen. John Tap. ‘The Pathway to Knowledge,’ &c. Octavo (O).

London, sixteen-thirteen. Richard Witt. ‘Arithmeticall questions, touching the Buying or Exchange of Annuities; Taking of Leases for Fines, or yearly Rent; Purchase of Fee-Simples; Dealing for present or future Possessions; and other Bargaines and Accounts, wherein allowance for disbursing or forbeareance of money is intended; Briefly resolved by means of certain Breviats.’ Quarto (octavo size) (O). [Pg 34]

There is also in the title-page ‘Examined also and corrected at the Presse, by the Author himselfe.’ A great many circumstances induce me to think that the general fashion of correcting the press by the author came in with the seventeenth century, or thereabouts.

Breviats are tables. As far as I know, this is the first English book of tables of compound interest. And there are real tables of half-yearly and quarterly compound interest. Again, decimal fractions are really used: the tables being constructed for ten million of pounds, seven figures have to be cut off, and the reduction to shillings and pence with a temporary decimal separation, is introduced when wanted. For instance, when the quarterly table of amounts of interest at ten per cent is used for three years, the principal being 100l. (page 99), in the table stands 137266429, which multiplied by 100 and seven places cut off gives the first line of the following citation:

‘The Worke
Facit		l	1372	66429
		sh	13	2858
		d	3	4296.’

Giving 1372l. 13s. 3d. for the answer. And the tables are expressly stated to consist of numerators, with 100 ... for a denominator.

[A reprint of this work, by T. Fisher, London, sixteen-thirty-four, duodecimo, is in the Royal Society’s Library.]

Paris, sixteen-fourteen. Artabasda. ‘Nic. Smyrnæi Artabasdæ Græci Mathematici ΕΚΦΡΑΣΙϹ Numerorum Notationis per gestum digitorum. Item Venerab. Bedæ de Indigitatione et manuali loquelæ lib.’ Octavo in twos (edited by F. Morell, the first Gr. Lat., the second Lat.).

Paris, sixteen-fourteen. Honorat Meynier. ‘L’Arithmetique ... enrichie de ce que les plus doctes mathematiciens ont inventés ... tant aux formes que nos anciens ont practiquees comme en celles qui se practiquent aujourd’huy en France, en Holande, en Allemagne, en Espagne et autres nations.’ Quarto (O). [Pg 35]

London, sixteen-fourteen. Thomas Bedwell. ‘De Numeris Geometricis. Of the nature and properties of geometrical numbers, first written by Lazarus Schonerus, and now Englished,’ &c. Quarto.

On figurate, square, &c. numbers, with applications to mensuration.

Bologna, sixteen-sixteen. P. A. Cataldi. ‘Quarta parte della pratica Aritmetica Dove si tratta della principalissima, et necessariissima Regola chiamata comunemente del Tre,’ &c. Folio (quarto size).

Edinburgh, sixteen-seventeen. John Napier. ‘Rabdologiæ seu Numerationis per Virgulas Libri duo: cum Appendice de expeditissimo Multiplicationis Promptuario, Quibus accessit et Arithmeticæ Localis Liber unus. Authore et inventore Johanne Nepero,’ &c. Duodecimo (O).

Oppenheym, sixteen-seventeen and nineteen. Robert Fludd. ‘Utriusque Cosmi Majoris scilicet et minoris metaphysica, physica atque technica historia.’ Two volumes. Folio in twos.

London, sixteen-nineteen. Henry Lyte. ‘The art of tens, or decimall arithmeticke.’ Octavo (duodecimo size).

One of the earliest English users of decimal fractions. P. 440.

Hamburgi, sixteen-twenty. Francis Brasser and Otto Wesellow. ‘Arithmetica Francisci Brasseri ... ab Ottone Wesellow ex Germanico in Latinum sermonem versa, atque in lucem edita.’ Octavo (O).

Hamburg, sixteen-twenty-one. Peter Lauremberg of Rostoch. ‘Institutiones Arithmeticæ.’ Octavo (small).

A demonstrative book.

Rostoch, sixteen-twenty-one. John Lauremberg. ‘Organum analogicum, sive instrumentum proportionum.’ Quarto.

This instrument is the sector.

Rostoch, sixteen-twenty-three. John Lauremberg. ‘Virgularum numeratricum et promptuarii arithmetici descriptio, figuræ et usus.’ Quarto.

A tract of fourteen pages on Napier’s rods.

London, sixteen-twenty-four. William Ingpen. ‘The Secrets of numbers, according to theologicall, arithmeticall, geometricall, and Harmonicall computation.’ Quarto.

A worthy follower of Peter Bungus. P. 425.

London, sixteen-twenty-four. Thomas Clay. ‘Briefe easie and necessary tables of interest and rents forborne,’ &c. Octavo (very small size).

[Pg 37] London, sixteen-twenty-eight. John Speidell. ‘An Arithmeticall Extraction or Collection of divers questions with their answers.’ Octavo (small).

Amsterdam, sixteen-twenty-nine. Albert Girard. ‘Invention nouvelle en l’algebre,’ &c. Quarto.

Oughtred’s Clavis is a book of arithmetic as well as of algebra, and one of great celebrity. The editions that have fallen in my way are:

London, sixteen-thirty-two. William Oughtred. ‘The Circles of Proportion and the Horizontal Instrument. Both invented and the uses of both Written in Latine by Mr. W. O. Translated into English: and set forth for the publique benefit by William Forster.’ Published with it, London, sixteen-thirty-three, ‘An addition unto the use of the instrument ... For the Working of Nauticall Questions ... Hereunto is also annexed the excellent Use of two Rulers for Calculation. And it is to follow after the 111 Page of the first Part.’ Quarto.

[Pg 39] London, sixteen-thirty-three. Robert Butler. ‘The Scale of Interest; or Proportionall Tables and Breviats ..... Together with the valuation of Annuities,’ &c. Octavo.

London, sixteen-thirty-three. Nicholas Hunt, M.A. ‘The Hand-Maid to Arithmetick refined: Shewing the variety and facility of working all Rules in whole Numbers and Fractions, after most pleasant and profitable waies. Abounding with Tables above 150. for Monies, Measures and Weights, tale and number of things here and in forraigne parts; verie usefull for all Gentlemen, Captaines, Gunners, Shopkeepers, Artificers, and Negotiators of all sorts: Rules for Commutation and Exchanges for Merchants and their Factors. A Table from 1l. to 100 thousand, for proportionall expences, and to reserve for Purchases.’ Octavo (ON). P. 442.

[Pg 40] Paris, sixteen-thirty-four (with another title-page, sixteen-forty-four, but it is all one impression). Peter Herigone. ‘Cursus Mathematici tomus secundus.’ Octavo (O).⁗

The volumes of this course, which is said to be the first complete course published, have different dates. It is a polyglott book, Latin and French. This second volume contains practical arithmetic. It introduces the decimal fractions of Stevinus, having a chapter ‘des nombres de la dixme.’ The mark of the decimal is made by marking the place in which the last figure comes. Thus when 137 livres 16 sous is to be taken for 23 years 7 months, the product of 1378′ and 23583‴ is found to be 32497374⁗ or 3249 liv. 14 sous, 8 deniers. There is a memoria technica for numbers in this book, which I subjoin, that those who know the old system of Grey may compare the two. The consonants and vowels which stand for numbers are:

P	B	C	D	T	F	G	L	M	N
1	2	3	4	5	6	7	8	9	0
a	e	i	o	u	ar	er	ir	or	ur
					ra	re	ri	ro	ru

So that r signifies five except before or after u, when it destroys the meaning altogether. Grey’s system is:

B	D	T	F	L	S	P	K	N	Z
1	2	3	4	5	6	7	8	9	0
a	e	i	o	u	au	oi	ei	ou	y

The vowel-systems of the two might be combined.

London, sixteen-thirty-four. William Webster. ‘Webster’s tables for simple interest direct .... Also his Tables for Compound Interest ...’ Octavo (small). Third edition.

London, sixteen-thirty-seven (again in forty-seven). Leonard Digges. ‘A book named Tectonicon .... Published by Leonard Digges, Gentleman, in the yeere of our Lord 1556.’ Quarto. [Pg 41]

London, sixteen-thirty-eight; reprinted in seventeen-forty-eight. John Penkethman. ‘Artachthos, or A New Book declaring The Assise or Weight of Bread by Troy and Avoirdupois Weights. Containing divers Orders and Articles made and set forth by the Right Honorable the Lords and Others of his Majesty’s most Honorable Privy Council.... Published by their Lordships Orders.’ Quarto.

[Leyden, sixteen-forty.] Adrian Metius. ‘Arithmetica Practica.’ Quarto (N).

There is no prolixity about this book. Sexagesimal fractions are taught, but not decimal ones. The title is torn out in my copy.

Herbornæ Nassoviorum, sixteen-forty-one. Joh. Henr. Alsted. ‘Methodus Admirandorum Mathematicorum Novem libris exhibens universam Mathesin.’ Duodecimo (N).

This has a slight treatise on arithmetic, with a few words on algebra: nothing on decimal fractions.

Sora, sixteen-forty-three. John Lauremberg. ‘Arithmetica ... exemplis historicis illustrata, itidem algebræ principia.’ Quarto (N).

A book which uses the Italian method of division is a rarity among German books of this period. The examples are drawn from historical matters, and from the military art.

Paris, sixteen-forty-four. Theon of Smyrna. ‘Τῶν κατὰ μαθηματικὴν χρησίμων εἰς τὴν τοῦ Πλάτωνος ἀνάγνωσιν.’ Quarto. Edited by Ismael Bullialdus (Bouillaud).

Leyden, eighteen-twenty-seven. Theon of Smyrna. The same title. Octavo in twos. Edited by J. J. de Gelder.

London, sixteen-forty-five. Edmund Wingate. ‘Use of the Rule of Proportion in Arithmetique and Geometric.’ Duodecimo. (A translation from Wingate’s own French.)

This book is on the use of Gunter’s logarithmic scale, not the sliding rule, which Wingate never wrote upon.

Leyden, sixteen-forty-six. Francis Vieta. ‘Opera Mathematica.’ Folio in twos.

London, sixteen-forty-eight. Seth Partridge. ‘Rabdologia, or the Art of numbring by Rods.’ Duodecimo (O).

Leyden, sixteen-forty-nine. Joh. Hen. Alsted. ‘Scientiarum omnium Encyclopædia.’ Two volumes. Folio in threes. [Pg 43]

London, sixteen-fifty. John Wybard. ‘Tactometria. seu, Tetagmenometria. Or, The Geometry of Regulars practically proposed ... by J. W.’ Octavo in twos.

An excellent book of mensuration of solids, full of remarkable information on the subject of weights and measures.

London, sixteen-fifty. Jonas Moore. ‘Moore’s Arithmetick: Discovering the secrets of that Art, in Numbers and Species.’ Octavo (N).

London, sixteen-sixty. ‘Moor’s Arithmetick in two books.’ Octavo.

Dantzig, sixteen-fifty-two. Joh. Broscius. ‘Apologia pro Aristotele et Euclide.... Additæ sunt duæ disceptationes de numeris perfectis.’ Quarto.

[Pg 44] London, sixteen-fifty-three. Noah Bridges. ‘Vulgar Arithmetique, explayning the Secrets of that Art, after a more exact and easie way than ever.’ Octavo (N).

London, sixteen-sixty-one. Noah Bridges. ‘Lux Mercatoria, Arithmetick Natural and Decimal,’ &c. Octavo (N).

London, [sixteen-fifty-four]. E[dmund] W[ingate]. ‘Ludus Mathematicus. Or, The mathematical Game: Explain Explaining the description, construction, and use of the Numericall Table of Proportion.’ Duodecimo. (Again in eighty-one.)

London, sixteen-fifty-five. Thos. Gibson. ‘Syntaxis Mathematica.’ Octavo.

Oxonii, sixteen-fifty-seven. John Wallis. ‘Mathesis Universalis: sive Arithmeticum Opus Integrum,’ &c. Quarto. [Pg 45]

Wurtzburg, sixteen-fifty-seven-and-eight. Gaspar Schott. ‘Magia Universalis Naturæ et Artis.’ Quarto (four parts, variously bound in volumes).

Leyden, sixteen-sixty. Vincent Leotaud. ‘Institutionum arithmeticarum libri quatuor, in quibus omnia quæ ad numeros simplices, fractos, radicales, et proportionales pertinent precepta clarissimis demon strationibus tum arithmeticis tum geometricis illustrata traduntur.’ Quarto (O).

This is a clear but heavy work, running to 700 pages.

Herbipoli (Wurtzburg), sixteen-sixty-one. Gaspar Schott. ‘Cursus Mathematicus.’ Folio in threes.

London, sixteen-sixty-four. James Hodder. ‘Hodder’s Arithmetick: or, That necessary Art made most easie.’ The third edition, much enlarged. Duodecimo (O).

Deventer, sixteen-sixty-seven. Joachim Camerarius. ‘Explicatio in duos libros Nicomachi Geraseni ... et notæ Samuelis Tennulii in Arithmeticam Jamblichi....’ Quarto. Also, (one publication, though different in place and date.)

Arnhem, sixteen-sixty-eight. Jamblichus. ‘In Nicomachi Geraseni Arithmeticam introductionem et de Fato.’ Quarto (Gr. Lat.).

London, sixteen-sixty-eight. John Newton. ‘The Scale of Interest: Or the Use of Decimal Fractions.’ Octavo.

This book is further described in ‘Tables,’ Penny Cyclop. Suppl.: it was expressly intended for a school-book, though it is a strange one for the time.

London, sixteen-sixty-eight. William Leybourn.

A
Platform		for		Purchasers
Guide				Builders
Mate				Measurers.

Octavo (O).

The first book is on interest, the second and third on building and mensuration of work.

Toulouse, sixteen-seventy. Diophantus. ‘Arithmeticorum libri sex.’ Folio in twos. [Pg 47]

London, sixteen-seventy-three. Sam. Morland. ‘The Description and Use of two Arithmetick Instruments, together With a Short Treatise explaining and Demonstrating the Ordinary Operations of Arithmetick, As likewise A Perpetual Almanac And several Useful Tables.’ Octavo (duodecimo size).

London, sixteen-seventy-three. John Kersey. ‘Mʳ Wingate’s Arithmetick, containing a plain and familiar method,’ &c. Octavo.

This is called the sixth edition of Wingate (N), one of the very best of the old writers. This edition has an appendix in augmentation of the commercial part, by Kersey, who says the first edition (which I have never seen) was published about sixteen-twenty-nine.

Amsterdam, sixteen-seventy-three. William Bartjen. ‘Verniende cyfferinge....’ Octavo (small) (O).

A book of a decidedly commercial character, and with good force of examples.

London, sixteen-seventy-four. John Mayne. ‘Socius Mercatoris: or the merchant’s companion, in three parts.’ Octavo (N).

The three parts are on arithmetic, vulgar and decimal, on interest, and on solids and cask-gauging: the second and third have two separate title-pages, dated sixteen-seventy-three. There is a little algebra.

London, sixteen-seventy-four. John Collins. ‘An Introduction to Merchants-Accompts.’ Folio in twos.

Book-keeping by double entry. Collins says that his first edition was in 1652, and his second (which was nearly all destroyed by the fire) in 1664-5.

London, sixteen-seventy-six. A. Forbes. ‘The Whole Body of Arithmetick Made Easie.’ Duodecimo (O).

Not a very easy system.

London, sixteen-seventy-seven. Michael Dary. ‘Interest epitomised, both compound and simple ... whereunto is added, a Short Appendix For the Solution of Adfected Equations in Numbers by Approachment: Performed by Logarithms.’ Quarto (octavo size) (N).

London, sixteen-seventy-eight. William Leybourn. ‘Arithmetick Vulgar, Decimal, Instrumental, Algebraical.’ Octavo. Fourth edition. (The four parts have separate titles and paging.) (ON).

I have not met with any earlier edition. This work seems to have been substantially inserted in the course of mathematics by Leybourn, presently mentioned.

London, sixteen-eighty. Thomas Lawson. ‘A Mite into the Treasury, being a Word to Artists, Especially to Heptatechnists.’ Quarto.

London, sixteen-eighty-one. Jonas Moore. ‘A new Systeme of the Mathematicks.’ Two volumes. Quarto.

The first volume contains arithmetic, much abridged from Moore’s larger work above noted. This course was written for Christ’s Hospital.

London, sixteen-eighty-two. Gilbert Clark. ‘Oughtredus [Pg 50] explicatus, sive Commentarius in Ejus Clavem Mathematicam.’ Octavo.

This commentary is now of no use. A wise commentator will endeavour to fill his book as much as he can with such little matters as will be historically interesting a couple of centuries after his own time; and this will in many cases cause him to outlive his author. I suppose this commentator is the Gilb. Clerke who published, in sixteen-eighty-seven, on what he called the Spot-dial.

London, sixteen-eighty-three. Peter Galtruchius. ‘Mathematicæ totius ... Clara, Brevis et accurata Institutio.’ Octavo.

There is a short treatise on arithmetic (O), and logarithms are mentioned as a recent invention, without any examples of their use.

London, sixteen-eighty-four. Rich. Dafforne. ‘The Merchant’s Mirrour.’ Folio in twos.

A book on book-keeping, translated from the Dutch.

London, sixteen-eighty-four. Abr. Liset. ‘Amphithalami or the Accountants closet.’ Folio.

London, sixteen-eighty-four. Wm. Leybourn. ‘The line of Proportion or [of?] Numbers, Commonly called Gunter’s Line made easie,’ &c. Duodecimo.

On the sliding rule, which invention is attributed to Seth Partridge. But he says that Milbourn (see Sherburne’s Manilius, or Penny Cycl. ‘Horrocks’) ‘disposed it in a Serpentine or Spiral Line;’ the logarithmic spiral, of course (Penn. Cycl. ‘Slide Rule’).

London, sixteen-eighty-four. Thos. Baker. ‘The Geometrical Key: or the Gate of Equations unlocked:’ &c. Quarto.

Polyglott: Latin and English. An attempt to do without arithmetic in the solution of equations, by means of construction. See the correspondence of Baker and Collins in the Macclesfield correspondence, beginning of vol. ii.

[Pg 51] London, sixteen-eighty-five. Seth Partridge. ‘The description and use of an instrument called the double scale of proportion.’ Octavo.

On the sliding rule.

London, sixteen-eighty-five. John Collins. ‘The doctrine of decimal arithmetick, simple interest, as also of compound interest ... made publick by T. D.’ Octavo (small).

A clear and short work, posthumous.

Naples, sixteen-eighty-seven. Ægid. Franc. de Gottignies. ‘Logistica Universalis.’ Folio in twos.

London, sixteen-eighty-seven. William Hunt. ‘The Gauger’s Magazine.’

A treatise on decimal fractions and mensuration, once well known. There is a table of squares in it up to that of 10,000.

London, sixteen-eighty-seven. D. Abercromby. ‘Academia Scientiarum..... Being a Short & Easie Introduction to the Knowledge Of the Liberal Arts and Sciences.’ Octavo (small).

A real smatterer’s book, published contemporaneously with the Principia of Newton. It gives names and general notions, with the names of a few celebrated authors; just enough to set up a man about town. There is another book, which may go with this, out of its place, being the earliest I have met with of the kind.

London, sixteen-forty-eight. Sir Balthazar Gerbier. ‘The interpreter of the Academie for forrain languages, and all noble sciences, and exercises.’ Quarto. [Pg 52]

London, sixteen-eighty-eight. Jonas Moore. ‘Moore’s Arithmetick: in Four Books.’ The third edition. Octavo.

A complete book of Arithmetic (N), with book III. on Logarithms, and book IV. on algebra, both dated sixteen-eighty-seven. This book was edited by John Hawkins, Cocker’s editor, who has put his name to the last three parts. See more of this presently, under Cocker.

London, sixteen-eighty-eight. Peter Halliman. ‘The Square and Cube Root compleated and made easie.’ Octavo (second edition) (O).

London, sixteen-ninety. Henry Coggeshall. ‘The Art of Practical Measuring Easily Performed, By a Two Foot Rule Which slides to a Foot,’ &c. Duodecimo in threes.

On the sliding rule and the use of logarithms.

London, sixteen-ninety. William Leybourn. ‘Cursus Mathematicus ... in nine books.’ Folio in twos and fours. [Pg 53]

The arithmetic (ON) Vulgar, Decimal, Instrumental, and Algebraical, is stated to be the first book: but algebra is afterwards promoted to follow geometry. Leybourn gives both methods of division, but thinks the old one not inferior to the new.

Leyden, sixteen-ninety. Claude Fr. Milliet Dechales. ‘Cursus seu Mundus Mathematicus.’ Folio in twos. Four volumes.

The first volume contains a course of Arithmetic (O), on which I may remark, as just now in the case of Schott, that it is very scanty. It is, perhaps, only in a collected course of Mathematics, that we see how difficult a thing computation must have been considered. And I begin to see perfectly, by instance after instance, that it was not the Greeks only who fled to geometry to escape from arithmetic.

Amsterdam, sixteen-ninety-two. Bernard Lamy. ‘Elemens des Mathematiques,’ &c. Duodecimo. Third edition (O).

The arithmetic is very poor: p. 410, a friend sends him the mode of using the carriage in subtraction; he having previously borrowed from the upper line: he prints this as a novelty.

London, sixteen-ninety-three. John Wing. ‘Heptarchia Mathematica ... Arithmatick....’ Octavo (N).

A plain book, containing, among other things, arithmetic, and a larger collection than usual of tables useful in mensuration. It has the licenser’s permission dated after the Revolution (Sept. 14, 1689).

Paris, sixteen-ninety-three. Nicolas Frenicle. In the ‘Divers Ouvrages de Mathématique et de Physique,’ folio, are contained the following tracts of Frenicle, namely, ‘Methode pour trouver la solution des Problemes par les exclusions—Abregé des combinaisons—Des Quarrez magiques—Table generale des Quarrez magiques de quatre de costé.’ And in another collection of Frenicle’s tracts only, quarto, no date nor place, with only fly title-pages (pp. 374, register A-Yy), there are the preceding tracts and also ‘Traité des Triangles Rectangles en nombres,’ in two parts, [of which last there is said to be an edition Paris, sixteen-seventy-six, octavo, which may be the one just mentioned, for aught I know].

Paris, sixteen-ninety-four. Jean Prestet. ‘Nouveaux Elemens de Mathematiques.’ Quarto (two volumes, third edition) (ON).

This work is a complete body of arithmetic and algebra, treated with depth and clearness, and embracing a great extent of subject. The author is much more impressed with the necessity of strict and sustained reasoning in algebra than most of his contemporaries; and in particular, his demonstrations of the processes of arithmetic are ample; a very uncommon thing in his day. Montucla, who does not appear to have known this third edition (for he mentions only the two former ones as of sixteen-seventy-five and eighty-nine), dismisses it with an opinion that it is ouvrage très estimable. He speaks of many inferior works with much higher praise.

London, sixteen-ninety-four. Wm. Leybourn. ‘Pleasure with Profit: Consisting of Recreations of Divers Kinds, viz. Numerical, &c.’ Folio in twos.

The recreations on arithmetic contain a great many of those which are still common. But some have gone out of vogue. I do not remember ever having had the pleasure or profit of the following:—To put together five odd numbers to make 20. Answered as follows:—Three nines turned upside down, and two units. Honest Leybourn thinks this answer is a Falacy; in which I differ from him: I think the question more than answered, viz. in very odd numbers.

London (no date, but about sixteen-ninety-five). Venterus Mandey. ‘Synopsis Mathematica Universalis: or the Universal Mathematical Synopsis of John James Heinlin, Prelate of Bebenhusan.’ Octavo.

London, sixteen-ninety-six. Samuel Jeake. ‘Λογιστικηλογία, or Arithmetick Surveighed and Reviewed.’ Folio (ON).

To see the size and weight of this book, one would have thought arithmetic had been a branch of controversial divinity. But it is now very valuable, from the variety of the information which it contains, particularly on weights, measures, and coins; and from the goodness of the index. There is a good deal of algebra in it, many quaint names, and stories of the same kind. Thus, in stating the famous story of the Delian oracle telling the inquirers to double a cube, in order to rid themselves of the plague, it is stated that the meaning was, that the best method to deliver realms from such contagion, was to abate of their voluptuousness, and apply themselves to literature. But for all this, those who know the value of a large book with a good index, will pick this one up when they can. P. 442.

Paris, sixteen-ninety-seven. De Lagny. ‘Nouveaux Elemens d’Arithmetique et d’Algebre.’ Octavo (duodecimo size).

He has the new method of division, but is obliged to write down the divisor afresh every time he wants to use it. As an instance of the effect of change of times and methods, the following trivial circumstance may be worth repeating. De Lagny says that it would take an ordinary computer a month to find the integers in the cube root of 696536483318640035073641037. To shew what would be thought of such an assertion now, I had put down the work for all the integers and a few decimals, for insertion in the Companion to the Almanac. When the time came for making up the manuscript, the slip of paper on which the above was written found itself (as the French say, which was more than I could do) in a heap of unarranged papers: and it was better worth my while to repeat De Lagny’s month’s work, than to sort the papers and pick it out.

Oxford, sixteen-ninety-eight. E. Wells. ‘Elementa arithmeticæ numerosæ et speciosæ.’ Quarto (octavo size) (N).

London, no date (but printed for W. and J. Marshal). Andrew Tacquet. ‘The Elements of Arithmetick In Three Books, The Seventh, Eighth and Ninth of Euclid: With the Practical Arithmetic In Two Books.... Translated into English by an Eminent Hand.’ Octavo.

Dr. Peacock mentions Tacquet as a late instance of the old method of division. Both methods are given in this translation, “whereof the first,” meaning the Italian method, “is least used but is the best; the other most used, but the difficultest.” [According to Dr. Peacock, the original is “Arithmeticæ Theoria et Praxis,” Antwerp, sixteen-fifty-six ——.] I have described an edition of Tacquet elsewhere (see the Index).

London, seventeen-hundred. Edward Cocker. ‘Cocker’s Arithmetick: Being A plain and familiar Method, suitable to the meanest Capacity for the full understanding of that Incomparable Art, as it is now taught by the ablest SchoolMasters in City and Country. Composed By Edward Cocker, late Practitioner in the Arts of Writing Arithmetick, and Engraving. Being that so long since promised to the World. Perused and Published by John Hawkins Writing-Master near St. Georges Church in Southwark, by the Authors correct Copy, and commended to the World by many eminent Mathematicians and Writing-Masters in and near London.’ Duodecimo.

London, sixteen-eighty-five. ‘Cocker’s Decimal Arithmetick ... Whereunto is added his Artificial Arithmetick ... also his Algebraical Arithmetick ... according to the Method used by Mr. John Kersey in his Incomparable Treatise of Algebra. Composed by Edw. Cocker ... Perused, Corrected, and Published by John Hawkins....’ Octavo.

London, seventeen-hundred. Christopher Sturmius. ‘Mathesis Enucleata, or the Elements of the Mathematicks.... Made English by J. R. M. and R. S. S.’ Octavo.

Arithmetic and Algebra are here very nearly separated.

Edinburgh, seventeen-one. George Brown. ‘A Compendious, but a Compleat System of Decimal Arithmetick, Containing more Exact Rules for ordering Infinites, than any hitherto extant.... First Course.’ Quarto.

The author’s knowledge of what was then extant, seems far from complete.

[Pg 63] London, seventeen-three. John Parsons and Thos. Wastell. ‘Clavis Arithmeticæ.’ Octavo (small) (ON).

The old system of division is rather recommended. There is a very neat work on algebra at the end.

Amsterdam, seventeen-four. Andrew Tacquet. ‘Arithmeticæ Theoria et Praxis.’ Octavo (small) (ON).

There had been several preceding editions. The theory consists in a version of the seventh, eighth, and ninth books of Euclid: the practice in an ordinary treatise. Both methods of division are given: the Italian as best, but least used.

London, seventeen-six. William Jones. ‘Synopsis Palmariorum Matheseos; or a New Introduction to the Mathematics.’ Octavo in twos (N).

Jones is well known among the contemporaries of Newton, and was the father of his celebrated namesake, the Indian Judge. The book, as its name imports, is a kind of syllabus.

London, seventeen-seven. ****, Art. Bac. Trin. Col. Dub. ‘Arithmetica absque Algebra aut Euclide demonstrate. Cui accesserunt, Cogitata nonnulla de Radicibus Surdis, de Æstu Aeris, de Ludo Algebraico, &c.’ Octavo in twos (NO).

There is no doubt that the author was the celebrated Bishop Berkeley, then a youth under twenty. The object of the arithmetic is stated in the title; and at that time the effort was much wanted. The algebraical game defies brief explanation.

London, seventeen-seven. John Smart. ‘Tables of simple interest and discount.’ Octavo in twos (small).

London, seventeen-fourteen. Samuel Cunn. ‘A New and Compleat Treatise of the Doctrine of Fractions, Vulgar and Decimal,’ &c. Octavo in twos (small) (N).

Prefixed is a testimonial from Halley, vouching for the goodness of the work and the novelty of some of its rules.

London, seventeen-fourteen. Edward Wells. ‘The young Gentleman’s Course of Mathematicks.’ Three volumes. Octavo.

These volumes (in my copy) exhibit what I have more often found at the beginning of the eighteenth century than at any other time; namely, volumes of different editions in one set. The Arithmetic title-page of the first volume, which follows the general title-page, has ‘seventeen-twenty-three,’ and ‘second edition.’ The other volumes have seventeen-fourteen and seventeen-eighteen. This Wells is the same whose work I have mentioned above; he qualifies himself D.D. and rector of Cotesbach in Leicestershire. When I quoted his diatribe against butter and cheese, ginger and pepper (which I did before I had seen this work), I sympathised with him, thinking he meant that liberal education had its wants as well as professional. But I was mistaken: it is gentlemanly education, as opposed to that of “the meaner part of mankind,” that he wants to provide for. Every page is headed on one side ‘The young gentleman’s,’ on the other ‘arithmetic,’ ‘geometry,’ or ‘mechanics,’ as the case may be. The gentlemen are those whom God has relieved from the necessity of working, for which he expects they should exercise the faculties of their minds to his greater glory. But they must not ‘be so Brisk and Airy, as to think, that the knowing how to cast Accompt is requisite only for such Underlings as Shopkeepers or Tradesmen;’ and, for the sake of taking care of themselves, ‘no Gentleman ought to think Arithmetick below Him, that do’s not think an Estate below Him.’ This Wells might be made as useful now as the Spartans used to make their slaves. The Arithmetic is an abridged version of the work of sixteen-ninety-eight above described.

London, seventeen-fourteen. Joh. Ayres. ‘Arithmetick Made Easie For the Use and Benefit of Trades-Men.’ Duodecimo. Twelfth edition; with an Appendix on Book-keeping by Chas. Snell.

A work of the immediate school of Cocker.

[Pg 65] London, seventeen-fifteen. John Hawkins. ‘Cocker’s English Dictionary.’ Second edition. Octavo in twos (small).

I have entered this book here only because Hawkins asserts that it was the work of the celebrated arithmetician: which I do not believe, for the reason above given. [The first edition is said to have the date seventeen-four.]

London, seventeen-seventeen. Wm. Hawney. ‘The Compleat Measurer....’ Duodecimo in threes.

A full treatise on decimal arithmetic.

‘Arithmetica Infinita, or the Accurate Accomptant’s Best Companion.’ Octavo in twos (small, oblong).

This is not a work on arithmetic, but a set of tables, which will certainly be reprinted as soon as the decimals of a pound gain their proper footing. The main part of it is the first nine multiples (and the 365th) of the decimals which express each farthing of the pound. Thus under 4s. 1¼d. are given the multiples of ·20520833..... The whole work is copperplate engraving from beginning to end. From several indications, I gather that Geo. Brown of this work is also the author mentioned under seventeen-one.

London, seventeen-seventeen. Roger Rea. ‘The Sector and Plain Scale, Compared ... Unto which is annexed, So much of Decimal Arithmatick and the Extraction of the square Root, as is necessary for the Working of Arithmetical Trigonometry.’ Octavo (N).

The treatise of an illiterate and confused person. Nothing has been more common than for those who write on application to consider it advisable not to trust the books to teach, nor the readers to know, decimal fractions, and to supply a fresh treatise. Rea says he uses the Italian mode of division (N) as being that which is most commonly used: nothing more than this, even in 1717.

London, seventeen-eighteen. Wm. Bridges. ‘An Essay to facilitate Vulgar Fractions; After a New Method, and to make Arithmetical Operations Very Concise &c. Duodecimo.’

London, seventeen-nineteen. Good. ‘Measuring made Easy.’ Octavo (duodecimo size).

A description of Coggeshall’s sliding rule, corrected and enlarged by James Atkinson.

London, seventeen-nineteen. John Ward. ‘The Young Mathematician’s Guide.’ The third edition. Octavo in twos (N). [Pg 66]

This useful course, which commences with arithmetic [was first published about seventeen-six]. It is recommended by Raphson and Ditton. The sixth edition was in seventeen-thirty-four; the eighth edition was in seventeen-forty-seven.

London, seventeen-twenty-one. Wm. Beverege (Bp. of St. Asaph). ‘Institutionum Chronologicarum Libri Duo. Unà cum totidem Arithmetices Chronologicæ Libellis.’ Third edition. Octavo in fours.

The date of the preface is sixteen-sixty-eight. The arithmetical part is a treatise on the numerals of different nations, learned, but not always judicious, according to modern views of the history of symbols. It is followed by a brief elementary treatise on arithmetic, with chronological examples.

London, seventeen-twenty-six. E. Hatton. ‘The Merchant’s Magazine: or, Trades-Man’s Treasury.’ Quarto.

Witemberg, seventeen-twenty-seven. Joh. Fr. Weidler. ‘De Characteribus Numerorum Vulgaribus et eorum Ætatibus ... Dissertatio Critico-mathematica.’ Quarto.

London, seventeen-twenty-eight. E. Hatton. ‘A Mathematical Manual: or Delightful Associate.’ Octavo.

Mostly on the use of the globes, but containing some “mysterious curiosities in numbers.”

London, seventeen-thirty. Alexander Malcolm. ‘A New System of Arithmetick, Theorical and Practical.’ Quarto.

One of the most extensive and erudite books of the last century, having 640 heavy quarto pages of small type; “wherein,” to go on with the title-page, “the science of numbers is demonstrated in a regular course from its first principles through all the parts and branches thereof, either known to the ancients or owing to the improvements of the moderns; the practice and application to the [Pg 67] affairs of life and commerce being also fully explained: so as to make the whole a complete system of theory, for the purposes of men of science; and of practice for men of business.” I quote this lengthy title as a true description of the work, at the date of publication. Probably the union of such masses of scientific and commercial arithmetic made the book unusable for either purpose.

London, seventeen-thirty-one. Edw. Hatton. ‘An Intire System of Arithmetic ... containing, I. Vulgar. II. Decimal. III. Duodecimal. IV. Sexagesimal. V. Political. VI. Logarithmical. VII. Lineal. VIII. Instrumental. IX. Algebraical. With the Arithmetic of Negatives and Approximation or [sic] Converging Series,’ &c. Second edition. Quarto.

A sound, elaborate, unreadable work, of 500 pages, of the same character as Malcolm’s.

London, seventeen-thirty-one. Wm. Hodgkin. ‘A Short New and Easy Method of Working the Rule of Practice in Arithmetick.’ Octavo in twos.

An author who chooses his own examples can write a short method on any rule: but the first example taken at hazard will probably defy the abbreviations.

London, seventeen-thirty-two. Joseph Champion. ‘Practical Arithmetick compleat.’ Octavo in twos.

London, seventeen-thirty-five. John Kirkby. ‘Arithmetical Institutions, containing a Compleat System of Arithmetic, Natural, Logarithmical, and Algebraical.’ Quarto in ones.

A system of arithmetic is mixed up with algebra. In the extraction of roots, Halley’s formula is applied in such a manner as to make the operation seem continuous, though it is just as difficult as before.

London, seventeen-thirty-five. James Lostau. ‘The Manual Mercantile, Second Book: Concerning Decimal Arithmetic....’ Quarto.

The first book was never published. This work contains a slight treatise on Arithmetic, but the body of it consists of all the various integers and fractions that may be useful in commerce, with the first nine multiples of each. It is 452 pages entirely of copperplate, the figures being rudely worked in, apparently by the author’s own hand. It is a posthumous work, and the editor says it took 17 years. This mode of stereotyping was adopted in several instances in the first half of the last century. And it must be observed, that if decimal arithmetic have not thriven in commercial affairs, it has not been for want of a great many attempts to facilitate the use of it, by publishing books of multiples.

[Pg 68] London, seventeen-thirty-five. Benjamin Martin. ‘A new Compleat and Universal System or Body of Decimal Arithmetick.’ Octavo in twos.

A very full system of decimal arithmetic, applied to all parts of commercial arithmetic.

London, seventeen-thirty-six. Thomas Weston. ‘A New and Compendious Treatise of Arithmetick.’ Quarto. Second edition.

A simple and useful treatise.

Edinburgh, seventeen-thirty-six. ——. ‘Arithmeticæ et Algebræ Compendium.’ Octavo in twos (N).

There is a small treatise on arithmetic. The publishers are Thos. and Wal. Ruddiman.

London, seventeen-thirty-eight. William Pardon. ‘A New and Compendious System of Practical Arithmetick.’ Octavo in twos.

Four hundred full octavo pages is not a very compendious book on arithmetic, or would not be so now: but it looked small by the side of Hatton, Malcolm, and Kirkby.

London, seventeen-thirty-eight. Tho. Everard. ‘Stereometry ... by the Help of a Sliding-Rule.’ Edited by Leadbetter, tenth edition. Duodecimo.

A book which once had a great reputation among excise collectors.

London, seventeen-thirty-nine. Christian Wolff. ‘A treatise of Algebra,’ translated from the Latin by J. H. M. A. Octavo.

There is very little of arithmetic, and that mostly on the properties of numbers.

Cambridge, seventeen-forty. Nicholas Saunderson. ‘The Elements of Algebra, in ten books.’ Quarto, two volumes.

This is a posthumous work of the well-known Professor Saunderson, the blind lecturer on optics. The first volume contains a synopsis of Arithmetic, and the editor’s account of the calculating board, by which Saunderson supplied the want of sight.

London, seventeen-forty. Wm. Webster. ‘Arithmetick in Epitome.’ Sixth edition. Duodecimo (ON). [Pg 69]

The eighth edition of the author’s book-keeping is London, seventeen-forty-four, octavo in twos (small); and the eighth edition of his ‘Attempt towards rendering the Education of Youth more easy and effectual,’ is London, seventeen-forty-three (with paging continued from that of the last). He says, “When a Man has tried all Shifts, and still failed, if he can but scratch out any thing like a fair Character, tho’ never so stiff and unnatural, and has got but Arithmetick enough in his Head to compute the Minutes in a Year, or the Inches in a Mile, he makes his last Recourse to a Garret, and, with the Painter’s Help, sets up for a Teacher of Writing and Arithmetick; where, by the Bait of low Prices, he perhaps gathers a Number of Scholars.”

London, seventeen-forty. ——. ‘A small Treatise of the Square and Cube....’ Second edition. Quarto (O). Also,

London, seventeen-forty. ‘A Supplement to the Square and Cube....’ Quarto in twos.

The author of this treatise (the last, I think, in which I have seen the old method of extracting the square root) is a copier of Peter Halliman, or some similar authority (see sixteen-eighty-eight), only his denominator is less by a unit.

London, seventeen-forty. Rob. Shirtcliffe. ‘The Theory and Practice of Gauging,’ &c.

A work once held in high estimation by the revenue-officers.

Edinburgh, seventeen-forty-one. John Wilson. ‘An introduction to Arithmetick.’ Octavo in twos.

A good demonstrative book, in a large type; very full on the complete operations with circulating decimals, the ignes fatui which have led many an arithmetical writer astray.

London, seventeen-forty-two. John Marsh. ‘Decimal Arithmetic made perfect.’ Quarto.

Almost entirely on infinite or circulating decimals. The predecessors whom he cites in his history of the subject are Wallis; Jones, 1706; Ward; Brown, 1708 or 1709 (he has not the work, but it is above at seventeen-one); Malcolm; Cunn; Wright, 1734; Martin, 1735; and Pardon. This subject of circulating decimals was at one time suffered to embarrass books of practical arithmetic, which need have no more to do with them than books on mensuration with the complete quadrature of the circle.

Leipzig, seventeen-forty-two. Jo. Christoph. Heilbronner. ‘Historia Matheseos Universæ.’ Quarto.

Though called a history of mathematics, and really a bibliography raisonnée, yet it is peculiarly devoted to arithmetic, the authors on which have a separate list. There are also dissertations on numerals, on their history, &c. The index of this book is of rare goodness.

[Pg 70] Geneva, seventeen-forty-three, forty-six, forty-seven, forty-nine, forty-one. Christian Wolff. ‘Elementa Matheseos Universæ.’ Quarto. A second or later edition (N). Five volumes.

The first of the five volumes contains a short treatise on arithmetic. Here, as happens so often in works of this period, a set is made up out of different editions. The first four volumes have editio novissima, the fifth has only nova. Wolff’s course would be better known if it were scarcer. The ordinary reader passes it by as an old book; the collector as one which is very common. But it is replete with pieces of information, which are historical references and suggestions. As far as I can remember, Wolff is much the most learned historian of those who have written extensive courses.

London, seventeen-forty-five. John Hill. ‘Arithmetick, Both in the Theory and Practice.’ Octavo in twos.

This is the seventh edition of a work of much celebrity. It seems to have owed its fame partly to a recommendation of Humphrey Ditton, prefixed to the first edition (about 1712), praising it in the strongest terms. Perhaps at this time the only things which would catch the eye are the table of logarithms at the end, and the powers of 2 up to the 144th, very useful for laying up grains of corn on the squares of a chess-board, ruining people by horseshoe bargains, and other approved problems.

London, seventeen-forty-eight. Charles Leadbetter. ‘The Young Mathematician’s Companion.’ Duodecimo in threes.

Second edition. Begins with an ordinary treatise on arithmetic.

London, seventeen-forty-eight. William Halfpenny. ‘Arithmetick and Measurement, Improved by Examples.’ Octavo in twos (N).

This is a surveyor’s and artisan’s book of application: but it contains decimal fractions.

London, seventeen-forty-nine; second part, seventeen-forty-eight. Solomon Lowe. ‘Arithmetic in two parts.’ Duodecimo in threes. [Pg 71]

London, seventeen-fifty. James Dodson. ‘The Accountant, or the method of Book-keeping Deduced from Clear Principles.’ Quarto.

As far as I can find, this is the first book in which double entry is applied to retail trade. James Dodson (my great-grand-father) is best known to mathematicians in general by his Antilogarithmic Canon, London, seventeen-forty-two, folio (see Penny Cycl. ‘Tables’).

London, seventeen-fifty. Daniel Fenning. ‘The Young Algebraist’s Companion.’ Duodecimo in threes.

There is a system of fractional arithmetic in this book, which is written in dialogue. The author thought it impossible to understand algebra without some better works on arithmetical fractions than then existed. As it is, says he, it is impossible to understand the Algorithm much less the Algorism, which he explains by saying that the former means the first principles, and the latter their practice. In this curious confusion of terms we see at its commencement an instance of a process which is always going on (though in this instance it has been arrested), the attachment of different meanings to different spellings of the same word. My curiosity led me to take a little trouble to trace Fenning to his authorities. And I find that of two writers who must have been in his hands, Saunderson and Kirkby, the first uses Algorithm for first principles, and the second Algorism for practical rules. I think I remember having seen a comparatively recent edition of this work.

London, ——. R. T. Heath, assisted by W. Davidson. ‘The Practical Arithmetician: or Art of Numbers improved.’ Duodecimo. (Revised by J. Bettesworth.)

Robert Heath was a person who made noise in his day, and in so doing established a claim to be considered a worthless vagabond. He was editor of the Ladies’ Diary from 1746 to 1753, when the Stationers’ Company found it absolutely necessary to strike out some of his scurrility, and dismiss him; appointing Thomas [Pg 72] Simpson in his placeBut before this, in 1749, Heath had commenced the Palladium, an annual publication resembling the Ladies’ Diary, the very first mathematical question of which is so expressed as to convey an indecent double-meaning, in a manner obviously intended. From 1750 or thereabouts, he began to write against Thomas Simpson. One of his publications against the latter is headed, “Miss Billingsgate in a salivation for a black eye:” and in a letter published in a newspaper, in 1751, he remarks of Simpson and another, that “the best writer against both is one who shall sign the warrant for their execution.”

London, seventeen-fifty-one. T. Smith. ‘Compendious Division. Containing, A Great Variety of Curious and Easy Contractions of Division.’ Octavo in twos.

London, seventeen-fifty-three. Sam. Stonehouse. ‘A Compendious Treatise of Arithmetic, By Way of Question and Answer.’ Octavo. Third edition.

Question and answer is well when the difficulty is in the question and the solution in the answer: but to turn “Decimals are divided like integers” into “Are decimals divided like integers? The division is exactly the same,” is trifling. This book gives the abbreviated rule for decimals of a pound, which very few books have given.

London, seventeen-fifty-six. John Playford. ‘Vade Mecum, or the Necessary Pocket Companion.’ Nineteenth edition. Quarto (long and narrow size).

This book is a ready reckoner, with miscellaneous tables. I have no information about the origin of these books, which I think are not so ancient as many may suppose. I could almost think from the preface (but such deductions are very deceptive) that the earliest of the books which are now called ready reckoners, meaning those which have totals at given prices ready cast up, was the following: London, sixteen-ninety-three. Wm. Leybourn. ‘Panarithmologia; Being A Mirror For Merchants, A Breviate For Bankers, A Treasure For Tradesmen, A Mate For Mechanicks, And A Sure Guide for Purchasers, Sellers, Or Mortgagers of Land, Leases, Annuities, Rents, Pensions, &c. In present Possession or Reversion. And A Constant Concomitant Fitted for All Men’s Occasions.’ Octavo.

London, seventeen-fifty-eight. Benjamin Donn. ‘A new Introduction to the Mathematicks, being Essays on Vulgar and Decimal Arithmetick. Containing, Not only the practical Rules, but also the Reasons and Demonstrations of them.’ Octavo (ON). [Pg 73]

This good book fulfils to a great extent the profession of the title-page as to demonstration. Donn had a good deal of miscellaneous information. He was, if I remember right, one of Humphrey Davy’s early teachers.

London, seventeen-fifty-nine and sixty-four. Benjamin Martin. ‘A New and Comprehensive System of Mathematical Institutions.’ Two volumes. Octavo in twos.

Old Ben Martin (as his admirers called him) was an able, and in this instance a concise writer. He wrote on every mathematical subject (and never otherwise than well, I believe, except on biography), and a complete set of his works is rarely seen. He was a bookseller.

London, no date, but about seventeen-sixty. William Weston. ‘Specimens of Abbreviated Numbers.’ Octavo in twos.

Some supposed new rules for formation and use of decimals.

London, seventeen-sixty. Jacob Welsh. ‘The Schoolmaster’s General Assistant.’ In two volumes. Octavo.

The author claims a hundred curious discoveries; what they are, I cannot find.

London, seventeen-sixty. James Dodson. ‘A Plain and Familiar Method for Attaining the Knowledge and Practice of Common Arithmetic.’ Octavo.

An edition of Wingate’s Arithmetic (N) called the nineteenth. But Wingate, who first published it about 1629, would not have known his own book, after the various dressings it received from Kersey, Shelley, and Dodson. One of George Shelley’s editions of John Kersey the son’s edition of John Kersey the father’s edition of Wingate, called the fourteenth, is London, seventeen-twenty, octavo in twos.

Amsterdam, seventeen-sixty-one. Isaac Newton. ‘Arithmetica Universalis.’ Quarto.

London, seventeen-sixty-five. J. Randall. ‘An Introduction To so much of the Arts and Sciences, More immediately concerned in an Excellent Education for Trade In its lower Scenes and more genteel Professions,’ &c. Duodecimo in threes.

Mr. Randall was a quaint man, but his book is well done. It contains arithmetic, mensuration, and geography; and ends with a dialogue between the heavenly bodies, upon their mutual arrangements, in which the earth insists upon being allowed to stand still, and quotes Scripture like an anti-Copernican, but is brought to reason by the arguments of the others. This is almost the only writer I have met with who has given the student a few hints upon habits of computation. Thus he will not let him say, three and four are seven, seven and five are twelve, &c.; but only three, seven, twelve, &c. For, says he, (the example being the addition of some rents) “as you have this pretty Income, you must talk like a Gentleman to your Figures.”

London, seventeen-sixty-six. W. Cockin. ‘A Rational and Practical Treatise of Arithmetic.’ Octavo in twos.

Dublin, seventeen-sixty-eight. Dan. Dowling. ‘Mercantile Arithmetic.’ Octavo in fours.

Mostly on exchanges. The author has given the rule for the instantaneous formation of the fourth and fifth and succeeding places of the decimal of a pound, which I never saw till now in any book but my own. The third edition of this book is Dublin, seventeen-ninety-five, duodecimo in threes. There is another author of the same name presently mentioned.

London, seventeen-seventy-one. Wm. Rivet. ‘An Attempt to illustrate the Usefulness of Decimal Arithmetic.’ Second edition. Octavo (small).

This book contains what I thought no one had given before myself, a complete head-rule for turning fractions of 1l. into decimals to any number of places. This method, which is much wanted in commercial arithmetic, is here lost amidst attempts to compute with interminable fractions; things on which real business will never waste a thought.

[Pg 75] Amsterdam, seventeen-seventy-three. Nicolas Barreme. ‘Comptes-faits, ou Tarif général des Monnoies.’ Duodecimo in threes.

This is a Dutch reprint of the work of a man who has given his name to ordinary mercantile computation in France, even more than Cocker in England. That he was a real person appears from the privilège copied into this book, dated Jan. 26, 1760, whereby Louis XV. grants the usual rights over his book to Nicolas Barreme. I have also seen in a catalogue an edition marked seventeen-forty-four. The name became an institution of France, which even the Revolution did not destroy. The Citoyen Blavier published a ‘Nouveau Barême,’ Paris, seventeen-ninety-eight, octavo, which he called a Barême décimal, in which there is a well-marked distinction between Barême the person and Barême the thing.

London, seventeen-seventy-three. Thomas Dilworth. ‘Miscellaneous Arithmetic.’ In seven parts. Duodecimo.

This work is but little known. Its contents are on the calendar; on logarithms; on the rule of three, when the first term is 1l. and all the terms are money; on the weather; a collection of riddles, answered, in the midst of which are seriously set forth Bacon’s paradoxes on the characteristics of a Christian, and an essay on the education of children. Dilworth had made his name a selling one, and was determined to make use of it.

Shrewsbury, seventeen-seventy-three. Thomas Sadler. ‘A Complete System of Practical Arithmetic ... on an entire new plan.’ Duodecimo in threes.

The newness of plan seems to consist in putting the rules into unintelligible verse, and beating even the older rule-mongers in puzzling plain questions. Thus, a cargo consists of 84, 61, and 35 tons, of which three-fifths is lost; what must each bear of the loss? This is done by first taking ⅗ of 180, namely 108; then 108 is divided by 180, producing ·6; then each of the parcels is multiplied by ·6.

London, seventeen-seventy-four. Anth. and Joh. Birks. ‘Arithmetical Collections and Improvements.’ Second edition. Octavo.

London, seventeen-seventy-four. Nich. Salomon. ‘The Expeditious Accountant; or, Cyphering rendered so short, That Half the Trouble.... A very curious WORK, Totally different from all that have preceded it.’ Octavo in twos. [Pg 76]

There is something new, says the author, in almost every rule: but I cannot find it. The head-rule for the decimals of a pound is introduced.

London, seventeen-seventy-seven. James Hardy. ‘The Elements or Theory of Arithmetic.’ Duodecimo in threes.

The author was a teacher at Eton, where, according to common notions, there could have been no such thing at the date above as a teacher of arithmetic. It is true there is an ambiguous comma in “Teacher of Mathematics, and writing-master at Eton College.” The book is a very creditable one, of great extent, including logarithms, &c.

London, seventeen-eighty. John Bonnycastle. ‘The Scholar’s Guide to Arithmetic.’ Duodecimo.

The first edition of this well-known work. It had from the beginning algebraical demonstrations attached.

Birmingham, seventeen-eighty-three. Wm. Taylor. ‘A Complete System of Practical Arithmetic.’ Octavo in twos.

An enormous book of 600 pages, with arithmetic, mensuration, geography, astronomy, algebra, book-keeping, &c., in the above order.

London, seventeen-eighty-four. G[eorge] Anderson. ‘The Arenarius of Archimedes ... from the Greek ... the dissertation of Christopher Clavius on the same subject....’ Octavo.

Oxford, eighteen-thirty-seven. Steph. Pet. Rigaud. ‘On the Arenarius of Archimedes.’ Octavo.

I choose this edition to introduce the only purely arithmetical work of Archimedes. Its object is to shew that any number, even a universe full of grains of sand, can be easily expressed: a thing by no means likely to be self-evident to a Greek, whose numerical notation was, till Archimedes and Apollonius shewed how it might be extended, far from sufficient for such a purpose. Professor Rigaud, one of the most learned and accurate of our modern inquirers into mathematical history, has given an account of Anderson, and many valuable remarks on his translation, as well as on the subject of it.

London, seventeen-eighty-four. Thomas Dilworth. ‘The Schoolmaster’s Assistant.’ Duodecimo.

This is the twenty-second edition. By the dates of the commendations prefixed, it would seem that the first edition was published in seventeen-forty-four or forty-five.

[Pg 77] Great-Yarmouth, seventeen-eighty-five. Thos. Sutton. ‘The Measurer’s Best Companion; or, Duodecimals brought to perfection.’ Octavo.

In truth this is the most elaborate system of duodecimals I have met with.

London, seventeen-eighty-six. George Atwood. ‘An Essay on the Arithmetic of Factors applied to various Computations which occur in the Practice of Numbers.’ Quarto.

Edinburgh, seventeen-eighty-six. John Mair. ‘Arithmetic, Rational and Practical.’ Octavo. Fourth edition.

The name of Mair, who was rector of the Perth Academy, is highly respected in Scotland. His ‘Book-keeping moderniz’d,’ of which the ninth edition is Edinburgh, eighteen-seven, octavo, had a great run. Completeness of subjects, and copiousness of examples, characterise both works, which extend to six hundred pages of small print each.

London, seventeen-eighty-seven. C. G. A. Baselli. ‘An Essay on Mathematical Language; or, an Introduction to the Mathematical Sciences.’ Octavo in twos.

It cannot be both, the reader will say: and in truth it is only the second, for arithmetic and algebra, with some good points about it.

London, seventeen-eighty-eight. ‘Clavis Campanologia, or a Key to the Art of Ringing.’ Duodecimo in threes.

As large a list as the present ought to have one book at least on bell-ringing, the whole theory of which is arithmetical. No art has had greater enthusiasts for it. The authors of the present treatise, Wm. Jones, Joh. Reeves, and Thos. Blakemore, are of the number. They record several names of inventors to whom they give words of praise which might apply to Newton or Euler; among them is Hardham, who is known to this day by the snuff mixture which he invented and sold in Fleet Street, where his name still remains. I should think that few of those whose noses he has tickled are aware that he may have done the same for their ears.

Oxford, seventeen-eighty-one (one volume); London, seventeen-eighty-eight (the other). James Williamson. ‘The Elements of Euclid, with dissertations....’ Quarto.

London, seventeen-eighty-eight. Thomas Keith. ‘The complete practical Arithmetician.’ First edition. Duodecimo in threes.

London, seventeen-ninety. Thomas Keith. ‘A Key to the complete practical Arithmetician.’ First edition. Duodecimo in threes.

London, no date; about seventeen-ninety (?). John Duncombe. ‘A new Arithmetical Dictionary.’ Octavo in twos.

The rules and terms of arithmetic, in alphabetical order.

Calcutta, seventeen-ninety. Joh. Thos. Hope. ‘A Compendium of Practical Arithmetick.’ Octavo in twos.

A clear, prolix book, for the Orphan School at Calcutta.

London, seventeen-ninety-one. William Emerson. ‘Cyclomathesis, or an easy Introduction to the several branches of the Mathematics.’ Octavo.

These are Emerson’s works, collected (not reprinted) in thirteen volumes, with new title-pages. The arithmetic, which is in the first volume, is said by the editor to be of seventeen-sixty-three. Emerson was the writer of many works, which had considerable celebrity: but he was as much overrated as Thomas Simpson was underrated. There is a most amusing life of him prefixed to the collection.

London, seventeen-ninety-one. Thos. Keith. ‘The New Schoolmaster’s Assistant.’ Duodecimo in threes.

An abridgment of the larger work above mentioned.

Berlin, seventeen-ninety-two. Leonard Euler. ‘L’Arithmétique raisonnée et démontrée, oeuvres posthumes de Léonard Euler, traduite en Francois par Bernoulli, Directeur de l’Observatoire de Berlin, &c. &c.’ Octavo in twos (ON). [Pg 79]

The editor calls this the first work of Euler in his preface, and posthumous in the title-page. It is, I suppose, a translation of the work which is set down in Fuss’s list as ‘Anleitung zur Arithmetic, 2 Th. Petersb. 1738. 8,’ the third of Euler’s separate works. It is mostly on commercial arithmetic, and shews that Euler did not, in 1738, consider the old method of division quite exploded.

London, seventeen-ninety-four. Rich. Carlile. ‘A Collection of one hundred and twenty ... Arithmetical, Mathematical, Algebraical and Paradoxical Questions.’ Octavo.

What Paradox is, as a science, I do not know: but the other distinctions are well known. All who know much of the country schools remember that mathematics meant geometry, as opposed to arithmetic and algebra. And it was right it should have been so: for neither the schoolboy’s arithmetic nor his algebra were disciplines.

London, seventeen-ninety-four. Henry Clarke. ‘The Rationale of Circulating Numbers.’ A new edition. Octavo.

Another tract on repeating decimals, with some additions on other subjects.

London, seventeen-ninety-four. Thomas Molineux. ‘The Scholar’s Question-book, or an introduction to Practical Arithmetic. Part the second. For the use of Macclesfield School.’ Duodecimo.

An ordinary school-book on fractions and commercial arithmetic. I never saw the first part. The school-seal, which is engraved on the title-page, gives the learner to understand the mode adopted of explaining difficulties; it displays a pedagogue with a birch-rod in his right hand and a book in his left; illustrative of primary and secondary method. The fourth edition of this second part is London, eighteen-twenty-two, duodecimo. As may be supposed from the date, the little hint does not appear.

Paris, seventeen-ninety-five. Agricol de Fortia. ‘Traité des Progressions ... précédé par un Discours sur la nécessité d’un Nouveau Systême de Calcul.’ Third edition. Octavo.

The new system of calculation is a proposal to annex to addition, multiplication, and involution, the next step, as the author takes it to be, in the chain of operations. But he is wrong in his use of the analogy. It appears that he was the author of several works on arithmetic.

Dublin, seventeen-ninety-six. John Gough (edited by his son). ‘Practical Arithmetick in four books.’ Duodecimo. [Pg 80]

This work, I am told, had such extensive currency in Ireland (where it was first published in seventeen-fifty-eight) that the name of the author became almost synonymous with arithmetic; insomuch that when Professor Thomson’s Arithmetic was first published in that country, it went by the name of ‘Thomson’s Gough.’ The second edition appears to have been an augmented and octavo work, afterwards reduced again for schools. It is a book of the ordinary character, and abounds in examples for practice. The last edition I have seen is Dublin, eighteen-thirty-one, duodecimo in threes, edited by M. Trotter.

New York, seventeen-ninety-seven. William Milns. ‘The American Accountant.’ Octavo in twos.

The author seems to have been an emigrant from St. Mary Hall, Oxford. His book has the peculiarity of giving in lieu of answers, the remainders to nine of the answers, for guide in the proof by casting out nines.

Paris, an VI. (seventeen-ninety-seven or ninety-eight). Condillac. ‘La Langue des Calculs.’ Octavo.

A posthumous work. The views of a clear-headed mathematician and metaphysician upon the foundation of arithmetic and the formation of its language.

London, seventeen-ninety-eight. Rich. Chappell. ‘The Universal Arithmetic.’ Octavo in twos (small).

London, seventeen-ninety-eight. Francis Walkingame. ‘The Tutor’s Assistant; being a Compendium of Arithmetic, and a complete question-book.’ Duodecimo in threes.

This is the twenty-eighth edition; when the first was published I do not know, any more than what edition is now current. I should be thankful to any one who would tell me who Walkingame was, and when the first edition was published: for this book is by far the most used of all the school-books, and deserves to stand high among them. I have before me John Fraser’s ‘Walkingame modernized and improved,’ London, eighteen-thirty-one, called seventy-first edition; John Little’s edition, London, eighteen-thirty-nine; William Birkin’s edition, Derby, eighteen-forty-three, called the fifty-first, and bearing proof that at least seven Birkins had appeared; and Samuel Maynard’s edition of F. Crosby’s edition, London, eighteen-forty-four. All [Pg 81] these are duodecimo. When editors do not agree within twenty as to the number of editions of their author which have been published, that author is surely a man of note.

London, seventeen-ninety-eight. Wm. Playfair. ‘Lineal Arithmetic, applied to shew the Progress of the Commerce and Revenue of England during the present century.’ Octavo in fours.

Not arithmetic, but plates arranging the several matters in curves, in the manner now much more familiar than it was then.

London, seventeen-ninety-nine. Charles Vyse. ‘The Tutor’s Guide, being a complete system of Arithmetic.’ Duodecimo in sixes. (Tenth edition, edited by J. Warburton; eleventh in eighteen-one.)

Paris, eighteen-hundred. ‘Séances des Ecoles Normales, recueillies par des sténographes, et revues par les Professeurs. Nouvelle édition.’ Thirteen volumes. Octavo.

Taunton, (no date, perhaps before eighteen-hundred). William Wallis. ‘An Essay on Arithmetic ... Briefly, Shewing, First, The Usefulness; Secondly, It’s extensiveness; Thirdly, The Methods of it.’

This is a remonstrance by the author, a teacher at Bridgwater in Somersetshire, against the prevailing modes of teaching arithmetic. The following is an extract: “And I have seen a Fair-Book (as ’tis call’d) of a young Man’s, about 17 Years of Age, who had been 6 Years at School, but never went through that Rule [of three]: In the same Book I found 132 Questions in Reduction, in the working of them were 2680 Figures, which might have been better done in 500, so that there were 2180 superfluous Ones. In another Rule I saw an Example, in which were 174 Figures, but might have been done in 23; and one of 80 that might have been done in 12: In general, I have found in the Boys Books, 3, or 4 Times as many Figures as need be. These Methods have so far hindred their Advances in Learning, that amongst 30 Scholars, since I came hither, I have not found one that understood a Rule beyond Division, tho’ some of them were 14 or 15 Years of Age, and had been kept at School, ever since they were capable of being taught.”

Paris, eighteen-hundred. Condorcet. ‘Moyen d’apprendre à compter surement et avec facilité.’ Second edition. Octavo (duodecimo size).

One of the simplest explanations of the most elementary arithmetic which has ever appeared. It was written in the last days of the author, while hiding from the fate which he only finally avoided by suicide: and the last sheet was hardly finished when his retreat was discovered.

Madrid, eighteen-one. ‘Aritmetica y Geometría práctica de la real Academia de San Fernando.’ Octavo.

A very clearly written and printed work.

[Pg 83] Buckingham, eighteen-two; second edition, eighteen-eight. John King. ‘An Essay intended to establish a new universal System of Arithmetic....’ Octavo.

The title of the second edition is more modest: it is ‘An Essay, or attempt towards establishing....’ The system is the octonary system, in which 10 means eight, 100 is sixty-four, &c.

London, eighteen-three. Rob. Goodacre. ‘Arithmetic adapted to different classes of learners.’ Duodecimo in sixes.

The ninth edition of this work, by Samuel Maynard, is London, eighteen-thirty-nine, duodecimo.

Dublin, eighteen-four. P. Deighan. ‘A Complete Treatise on Arithmetic, rational and practical.’ Two volumes. Octavo in twos.

London, eighteen-five. Christ. Dubost. ‘Commercial Arithmetic, with an Appendix upon Algebraical Equations.’ Duodecimo in threes.

Of all works I know professing to be strictly commercial, this has the fullest explanations in words of the rules and processes.

[Pg 84] Madras, eighteen-six. James Brown. ‘A course of Military and Commercial Arithmetic.’ Small octavo size: no signatures.

As might be expected, this is full on Indian exchanges, weights, and measures.

London, eighteen-six. William Frend. ‘Tangible Arithmetic; or the Art of Numbering made easy, by means of an Arithmetical Toy which will express every number up to 16,666,665.’ Octavo. Second edition.

The toy is the Chinese instrument or abacus, called the Schwanpan, for a description of which see Peacock, p. 408.

Paris, eighteen-seven. J. B. V. ‘L’Arithmétique enseignée par des moyens clairs et simples.’ Octavo.

This is in dialogue between a mother and her boy: the author assures us that they are from real life; and it must have been so; for the ancien officier du génie, as he calls himself, could no more have written these dialogues than the mother and child could have constructed a sap. The lady has the awkward name of Madame Épinogy; and as the object of the dialogues is to make the child invent, I can find no origin for this name, except the supposition that it is a blundering derivative from ἐπίνοια. Nevertheless the dialogues are exceedingly good.

Paris, eighteen-eight. F. Peyrard. ‘Oeuvres d’Archimède, traduites littéralement, avec un commentaire.’ Second edition, two volumes. Octavo.

London, eighteen-ten. W. Tate. ‘A System of Commercial Arithmetic.’ Duodecimo.

A work approximating more nearly to modern business than most of those then in use, in its additions; but, like most attempts to improve real commercial arithmetic, wanting the corresponding omissions.

Hawick, eighteen-eleven. Chas. Hutton, edited by Alex. Ingram. ‘A Complete Treatise on Practical Arithmetic and Book-keeping.’ Duodecimo in threes.

According to Hutton’s Catalogue, the fifth edition was in seventeen-seventy-eight, and the twelfth in eighteen-six: and at his death he possessed no edition previous to the fifth. The late Dr. Olinthus Gregory published what he called the eighteenth edition, enlarged, &c. London, eighteen-thirty-four, duodecimo: and a new edition of Ingram’s Hutton, by James Trotter, appeared Edinburgh, eighteen-thirty-seven, duodecimo.

London, eighteen-twelve. Thomas Clark. ‘A New System of Arithmetic; including Specimens of a Method by which most Arithmetical Operations may be performed without a Knowledge of the Rule of Three; and followed by Strictures on the Nature of the Elementary Instruction contained in English Treatises on that Science.’ Octavo.

This is an able attempt to draw public attention to the state of instruction in arithmetic. The author asserts, 1. There is not in the English language, a work of any repute whatever, employed in school education, in which the four fundamental rules of arithmetic are clearly and comprehensively laid down. 2. Not one in which the rules laid down are accompanied by examples so detailed as to remove the difficulties which these rules must present to beginners. 3. None in which the rules and examples for abstract and concrete numbers are kept distinct from each other. 4. There is not a work of this description in which ordinary and decimal fractions are properly arranged. 5. Or in which the rationale of arithmetical operations seems of sufficient importance to the instructor to induce him to incorporate it with his work. 6. Or in which the principles and algebraical signs used in arithmetic are given and explained at the time when the science requires their introduction.

[Pg 86] Dublin, eighteen-twelve. John Walker. ‘The Philosophy of Arithmetic ... and the Elements of Algebra.’ Octavo.

Mr. Walker was a good scholar, an excellent mathematician, and a most original thinker. Both this work and that which he published on geometry shew great power.

Sheffield, eighteen-thirteen. Joseph Youle. ‘The Arithmetical Preceptor ... to which is added a Treatise on Magic Squares.’ Duodecimo in threes.

London, eighteen-thirteen. Edward Strachey. ‘Biga Ganita; or the Algebra of the Hindus.’ Quarto.

Bombay, eighteen-sixteen. John Taylor. ‘Lilawati; or a Treatise on Arithmetic and Geometry by Bhascara Acharya.’ Quarto size (no signatures).

London, eighteen-seventeen. Henry Thomas Colebrooke. ‘Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara.’ Quarto.

The first work has notes by S. Davis, and is from a Persian version of Bhascara’s Sanscrit. The second work is also from the Persian. The third, which contains not only the two works of Bhascara, but also an arithmetical chapter from Brahmegupta, is all from the Sanscrit. It is also pretty copious in selections from the Commentators, and has a large body of dissertation by Colebrooke himself. But it does not entirely supersede the former two, which have likewise valuable annotations.

Edinburgh, eighteen-thirteen. Elias Johnston. ‘A sure and easy Method of learning to Calculate.’ Duodecimo in sixes.

This is a translation of the work of Condorcet mentioned under the date 1800.

Paris, eighteen-thirteen. F. Peyrard. ‘Les Principes fondamentaux de l’Arithmétique.’ Octavo in twos.

An elegant mixture of arithmetic and algebra, by the editor and translator of Euclid, and the translator of Archimedes.

Lille, eighteen-fourteen. ——. ‘Manuel d’Arithmétique ancienne et décimale.’ Duodecimo in threes.

A small book, in question and answer: a transition book from the old system to the new, containing both, and intended for commercial purposes. At the end are some forms for letters of ceremony and business, and for petitions: and it seems rather strange to English eyes to see that a petition for a son condemned to death for homicide, ranks among the matters which are considered near enough to the ordinary course of business to find a place; and a place which, when opened, gives the option of reading the way of turning francs into roubles.

[Pg 87] London, eighteen-fourteen. S. F. Lacroix. ‘Traité Élémentaire d’Arithmétique à l’usage de l’école centrale des quatre-nations.’ Tenth edition. Octavo.

Dublin, eighteen-fourteen. R. F. Purdon. ‘Theory of some of the Elementary Operations in Arithmetic and Algebra.’ Octavo.

Oxford, eighteen-fourteen. Charles Butler. ‘An Easy Introduction to the Mathematics.’ Octavo.

This book fulfils the promise of the title-page well, and has been frequently cited for the historical introductions to the several subjects, which are very good, and, as parts of a learner’s course, unexampled.

London, eighteen-fifteen. J. Carver. ‘The Master’s and Pupil’s Assistant.’ Duodecimo in threes.

The author of this work, dependent as the sale of it was on teachers, has had the sense and courage to say, that questions with answers are for the benefit of the masters and the injury of the pupils. It is dated from Belgrave House, Pimlico,—a name and site which might puzzle an antiquary a century hence.

Paris, eighteen-sixteen. Bezout. ‘Traité d’Arithmétique.’ Eighth edition. Octavo.

A work of a somewhat older stamp than those of Lacroix and Bourdon. The eleventh edition was Paris, eighteen-twenty-three, octavo, edited by A. A. L. Reynaud, with notes and a table of logarithms; the notes a separate work, with another title-page. The next year appeared the twelfth edition of Reynaud’s own work, Paris, eighteen-twenty-four, ‘Traité d’Arithmétique,’ octavo, also augmented by a table of logarithms. This work enters rather more on the theory of numbers.

London, eighteen-sixteen. Thos. Taylor. ‘Theoretic Arithmetic, in three books; containing the substance of all that has been written on this subject by Theo of Smyrna, Nicomachus, Jamblichus, and Boetius. Together with some remarkable particulars respecting perfect, amicable, and other numbers, which are not to be found in the [Pg 88] writings of any ancient or modern mathematicians. Likewise, a specimen of the manner in which the Pythagoreans philosophised about numbers; and a development of their mystical and theological arithmetic.’ Octavo.

Edinburgh, circa eighteen-sixteen. A. Melrose (edited by A. Ingram). ‘A Concise System of Practical Arithmetic.’ Second edition. Duodecimo in threes.

Horner refers to this book as containing what is nearly an anticipation of his method, in the case of the simple cube root.

London, eighteen-seventeen. Thos. Preston. ‘A New System of Commercial Arithmetic ... a perfect, a permanent and universal ready reckoner.’ Duodecimo in threes.

Perhaps the plan of this book is partly taken from one published by Girtanner in seventeen-ninety-four (Penny Cycl. Suppl. Tables), in which the logarithms of numbers and certain intermediate fractions are given. But, in the main, it is an application of the same principle as that which has long been used in astronomical logarithms, namely, giving the logarithms of integers, with a column in which those integers, considered as seconds, are turned into degrees, minutes, and seconds. In this way are given the logarithms of pence up to 130l.; of pounds up to 7 cwt. 16lbs.; of twelfths up to 333⅓; of sixteenths up to 312½; of sixteenths of gallons, considered as fractions of a tun of 236 gallons, for seed-oils; the same for a tun of 256 gallons, for fish-oils and wines; of pounds of 120 to the cwt. up to five tons; of grains, up to 25 oz. troy; and two for days and for pounds at 5 per cent, by which one operation gives the interest for days on any number of pounds up to 2600l.

Leipsic, eighteen-seventeen. Fred. Astius. ‘Theologumena Arithmeticæ.... Accedit Nicomachi Geraseni Institutio Arithmetica.’ Octavo. (See p. 17.)

These θεολογούμενα have been attributed to Jamblichus and to Nicomachus: but they seem rather to consist of extracts from Nicomachus, Anatolius, and others. They are explanations of the Pythagorean and Platonic opinions on numbers; and form a very good accompaniment for the works of Nicomachus. The notes are full and good.

London, eighteen-eighteen. George G. Carey. ‘A Complete System of Theoretical and Mercantile Arithmetic.’ Octavo.

A commercial book with a table of logarithms in it is rare in the nineteenth century.

Edinburgh, eighteen-eighteen. William Ritchie. ‘A System of Arithmetic.... and a Course of Book-keeping.’ Duodecimo in sixes. [Pg 89]

A book of much greater merit than could be guessed from its pretensions or its notoriety: and, for its size, one of the most comprehensive I have met with. Its author was a little (about twenty years) in advance of his age, and the greater part of the edition was sold as waste paper.

London, eighteen-nineteen. ‘Philosophical Transactions.’ Quarto. W. G. Horner. ‘A New Method of Solving Numerical Equations of all orders by Continuous Approximation.’

London, eighteen-thirty. Thos. Leybourn. ‘New Series of the Mathematical Repository.’ Volume five. Octavo in twos. W. G. Horner. ‘Horæ Arithmeticæ.’

Edinburgh, eighteen-twenty. John Leslie. ‘The Philosophy of Arithmetic.’ Octavo.

I have spoken of this work in the Introduction. P. 373, 405, 411, 477.

Vienna, eighteen-twenty-one. Geo. Fred. Vega. ‘Vorlesungen über die Rechenkunst und Algebra.’ Octavo.

The fourth edition (with preface dated seventeen-eighty-two, which I presume to be the date of the first) of the Arithmetic of the celebrated editor of the greatest modern table of logarithms.

[Pg 90] Leeds, eighteen-twenty-three. [Walker?] ‘Elements of Arithmetic for the use of the Grammar School....’ Duodecimo in threes.

An excellent little work, which I suppose I am right in attributing to Mr. Walker, and calling it the first edition of the work presently mentioned.

London, eighteen-twenty-three. Thomas Taylor. ‘The Elements of a New Arithmetical Notation, and of a New Arithmetic of Infinites.’ Octavo in twos.

A curious attempt at establishing a theory of infinites, the unit of which is 1 + 1 + 1 + &c. ad inf. Those who know nothing of Taylor the Platonist, should read his life in the Penny Cyclopædia. To re-establish Plato and Aristotle (in some sense even their very mythology) was the uniform endeavour of a long life and a most voluminous course of authorship. P. 424.

London, eighteen-twenty-four. Jas. Darnell. ‘Essentials of Arithmetic, or Universal Chain.’ Duodecimo.

This treatise reduces most questions to the form known as the chain-rule. Several writers have since advocated this plan.

London, eighteen-twenty-five. J. Joyce. ‘A System of Practical Arithmetic.’ Duodecimo.

A well-known book. The preface is dated eighteen-sixteen, which I suppose to be the date of the first edition.

London, eighteen-twenty-six. J. R. Young. ‘An Elementary Treatise on Algebra.’ Octavo.

I enter this work here as that of the first elementary writer who saw the value of Horner’s (or, as he then called it, Holdred’s) method of solving equations. This is, I believe, the first of the series of widely-known and much-used works by the same author.

London, eighteen-twenty-six. Chas. Pritchard. ‘Illustrations of Theoretical Arithmetic.’ Duodecimo in sixes.

A demonstrated system of Arithmetic, at a time when there were few such things in English.

London, eighteen-twenty-seven. H. and J. Grey. ‘Practical Arithmetic.’ Eighth edition. Duodecimo.

A book of concise rules for special cases.

London, eighteen-twenty-seven. George Walker (Master of the Grammar School at Leeds). ‘Elements of Arithmetic, theoretical and practical, for the use of the Grammar School, Leeds.’ Third edition. Duodecimo. [Pg 91]

A clear and excellent work, written by a man of real science. I doubt whether the peculiarity of the work, the introduction of the distinction of integers and decimal fractions at the very outset, be judicious: but it has had advocates of powerful name.

Paris, eighteen-twenty-eight. Bourdon. ‘Élémens d’Arithmétique.’ Sixth edition. Octavo.

More complete than Lacroix in details. Bourdon was an excellent elementary writer, both on arithmetic and on algebra. I began my career as an author, by a translation of part of his work on algebra.

London, eighteen-twenty-eight. John Bonnycastle. ‘The Scholar’s Guide to Arithmetic ... enlarged and improved by the Rev. E. C. Tyson.’ Duodecimo.

What edition this is I do not know.

London, eighteen-twenty-eight. ‘An Abridgment of the Arithmetical Grammar ... By Catechetical Scrutiny.’

I have never met with the larger work.

Newcastle, eighteen-twenty-eight. William Tinwell (edited by James Charlton). ‘Treatise on Practical Arithmetic, with Book-keeping, by single and double entry.’ Twelfth edition. Duodecimo in threes.

Dublin, eighteen-twenty-eight. John Garrett. ‘An Essay on Proportion.’ Octavo.

Mostly arithmetical, with something on the connexion of number and magnitude.

London, —— George Peacock. ‘Arithmetic.’

London, [eighteen-twenty-nine]. James Parker. ‘Arithmetic and Algebra.’ Octavo.

An early work of the Society for the Diffusion of Useful Knowledge. Arithmetical and algebraical demonstrations are connected.

London, eighteen-thirty. Joh. White. ‘An Elucidation of the Tutor’s Expeditious Assistant.’ Duodecimo.

The author claims as a new discovery in arithmetic, the notion of forming examples in which the figures of the answer have some perceptible relation, unknown to the pupil, but known to the teacher, who can thereby see whether it is right or wrong. All the questions set, for instance, in multiplication have digits which descend by twos, as in 86420, 20864, 75319, &c. So that it is gravely proposed that the pupil shall be trained upon selected questions in which the answers have all the figures odd, or all even. A pupil so trained would be in danger of being led to think that 3728 could not be the product of any numbers.

Edinburgh, eighteen-thirty. Adam Anderson. ‘Arithmetic.’ Quarto.

This is the article so headed in Brewster’s Cyclopædia, vol. ii. part 1. It has a fair amount of arithmetical history, though hardly enough for the extent of the work of which it forms a part.

London, eighteen-thirty-one. Frederic Rosen. ‘The Algebra of Mohammed Ben Musa.’ Octavo in twos. English and Persian.

This algebra is very arithmetical. It was published by the Oriental Translation Fund. M. Libri, in the work presently mentioned (vol. i. pp. 253-297), has given a long extract from old Latin translations now in manuscript in the Royal Library at Paris. This book passes for the one by means of which Leonard of Pisa was the first European who learnt algebra, or at least who wrote upon it.

Derby, eighteen-thirty-three. Samuel Young. ‘A System of Practical Arithmetic.’ Duodecimo.

A work with great force of rules, and examples in machinery, manufactures, &c.

[Pg 94] Springfield (U. S.), eighteen-thirty-three. Zerah Colburn. ‘A Memoir of Zerah Colburn, written by himself ... with his peculiar methods of calculation.’ Duodecimo in threes.

A great many will remember that in 1812 and 1815 two young boys, Zerah Colburn and George Bidder, astonished every one by a power of rapid mental calculation to which the most practised arithmeticians could not make the least approach. Mr. Colburn was, in 1833, a minister among the Methodists in the United States; Mr. Bidder, there is little occasion to say, is a civil engineer in England. The peculiarity of Colburn was, that he could extract roots and find the factors of numbers, to an extent which the mathematician himself had no organised rule for doing. Speaking in the third person, Mr. Colburn says: “Some time in 1818, Zerah was invited to a certain place, where he found a number of persons questioning the Devonshire boy (Mr. Bidder). He displayed great strength and power of mind in the higher branches of arithmetic; he could answer some questions that the American would not like to undertake; but he was unable to extract the roots and find the factors of numbers.” This treatise contains an account of Colburn’s methods.

London, eighteen-thirty-seven. Daniel Harrison. ‘A New System of Mental Arithmetic, by the acquirement of which all numerical questions may be promptly answered without recourse to pen or pencil ... an entirely novel method of reducing the largest sums of money ... to their lowest denominations by means of original quadrantal rationale’.... Duodecimo in threes. Second edition.

[Pg 95] Berwick-upon-Tweed, eighteen-thirty-eight. James Gray (edited by Wm. Rutherford). ‘An Introduction to Arithmetic.’ Duodecimo in threes.

Mr. Rutherford says this neat little work went through more than forty editions in the half century preceding this publication. How many works on arithmetic there must be which I have never seen! I never met with any one of the forty.

Paris, eighteen-thirty-eight, thirty-eight, forty, forty-one. Guillaume Libri. ‘Histoire des Sciences Mathématiques en Italie, depuis la renaissance des lettres jusqu’à la fin du xviiᵉ siècle.’ Octavo.

The first volume of this work was printed Paris, eighteen-thirty-five, octavo, but it was never sold; the whole impression (except a few copies distributed as presents) was burnt. The work is yet unfinished; four volumes, dated as above, being all that have appeared. It is a valuable history as concerns arithmetic, both in text and notes; the latter contain—the liber augmenti et diminutionis compiled by Abraham (supposed to be Aben Ezra) secundum librum qui Indorum dictus est (vol. i. pp. 304-376)—Account of an extract from the Liber Abbaci of Leonard of Pisa, written in twelve-hundred-and-two (vol. ii. pp. 287-304)—the practica Geometriæ of the same author, being the whole of his algebra, written in twelve-hundred-and-twenty (vol. ii. pp. 305-476)—extracts from Pacioli (vol. iii. pp. 277-294): besides many other matters not relating to arithmetic. As this work stands, it can be little used for want of an index.

London, eighteen-thirty-eight. Thos. Keith. ‘The Complete Practical Arithmetician.’ Duodecimo.

The editor (Mr. Maynard, the mathematical bookseller, who ought to know) calls this the 12th edition: it has a deservedly high character among books of rules only, for precision and completeness. Keith says that the bare names of those who have written on arithmetic in England, from the time of Wingate, would fill a moderate volume. I suspect not: after having examined every source within my reach, and got only 1500 names, out of all times and countries, I should think it impossible that Keith could have known of 300 Englishmen, within the limits mentioned.

London, eighteen-forty. Thomas Stephens Davies. ‘Solutions of the Principal Questions of Dr. Hutton’s Course of Mathematics.’ Octavo.

This work is also a running comment on the original, and it has a specific right to be in this list from its abounding with instances of Mr. Horner’s various methods of manipulating arithmetical and more particularly algebraic questions, independently of his celebrated method of solving equations. In eighteen-forty-one the same author published an edition of Hutton’s Course itself.

[Pg 96] London, eighteen-forty-two. ‘Encyclopædia Britannica.’ Vol. III. Quarto.

The article on arithmetic was written, I suppose, by Leslie; or if not, by a follower of his views. It is meagre in its history.

London, eighteen-forty-three. Alfr. Crowquill (as he calls himself). ‘The Tutors Assistant, or Comic Figures of Arithmetic.’ Duodecimo in sixes.

Walkingame’s arithmetic, with comic woodcuts: but the subject will not furnish good materials. Under ‘Subtraction of time,’ for instance, is represented the stealing of a watch; as a heading for Troy weight, the wooden horse; and for measure of capacity, a phrenologised head. The worst of it is, that the joke will remain on hand too long for the learner: a picture of a gamekeeper producing a hare from a poacher’s pocket must stare him in the face, as an illustration of ‘proof,’ through twenty-two mortal questions of addition. A comic arithmetic, with the cuts in illustration of the examples for exercise, would give the artist much fairer play.

Liège, eighteen-forty-four. H. Forir. ‘Essai d’un cours de Mathématiques à l’usage des élèves du collège communal de Liège ... Arithmétique.’ Eighth edition. Octavo in fours.

A good and well-printed treatise, with many examples.

London, eighteen-forty-five. Thomas Tate. ‘A Treatise on the First Principles of Arithmetic.’ Duodecimo.

Two pages at the end, on the use of the properties of nine in constructing questions for pupils, are well worthy the attention of teachers.

London, eighteen-forty-six. A. De Morgan. ‘The Elements of Arithmetic.’ Fifth edition, with Appendixes. Duodecimo in threes.

The previous editions were eighteen-thirty, thirty-two, and thirty-five, duodecimo, and eighteen-forty in threes.

Books of bibliography last longer than elementary works; so that I have a chance of standing in a list to be made two centuries hence, which the book itself would certainly not procure me.

[Pg 97]

The following are some additional works more briefly described. Here 2 + 2, 4 + 4, 6 + 6, mean quarto in ones, octavo in twos, duodecimo in threes.

No place, date, nor author’s nor printer’s name. ‘Algorithmus Integrorum Cum Probis annexis.’ Quarto in threes.

I think this is the oldest book in my list. The type is of that manuscript appearance which is so common in books printed before 1480: and this one can hardly be later than 1475, and may be ten years earlier. The matter is that of a writer prior to the school of Borgi and Pacioli. Though the nine digits are given, they are not used. The rules only are given, and they are Sacrobosco’s set; for which see p. 15.

No place nor date. ‘Algorismus novus de integris compendiose sine figurarum (more Italorum) deletione compilatus. artem numerandi omnemque viam calculandi enucleatim brevissimè edocens. una cum algorismis de minuciis vulgaribus videlicet et phisicalibus. Addita regula proportionum tam de integris quam fractis que vulgo mercatorum regula dicitur. Quibus habitis qui vis modica adhibita diligentia omnem calculandi modum facillime adipisci potest.’ Quarto in fours.

Three different editions of this work are mentioned by bibliographers, all without dates. Numeration is called prima species; addition, secunda species; and so on. There are slight examples, except upon the square root, in which only results are given. It is something between such a work as the last, with rules only, and that of Borgi, to which I now come, in which there is decided force of examples.

Venice, fourteen-eighty-eight. Piero Borgi. Work on Arithmetic without title, as follows. Quarto in fours.

Paris, fourteen-ninety-six. ‘In hoc opere contenta. Arithmetica decem libris demonstrata Musica libris demonstrata quatuor Epitome in libros arithmeticos divi Severini Boetii. Rithmimachie ludus qui et pugna numerorum dicitur.’ Folio in fours.

This is the first edition of the work mentioned in page 10: so that I was wrong in page 4. I was misled by bibliographers mentioning this as an edition of Jordanus only. The copy before me comes to me in two different parts, though the register at the end (which has the first words of all the sheets) proves that these parts belong to one work. It would be odd if it should have been customary to divide it in this manner, so as to lead to the above imperfect description. The corrector of the press was David Lauxius, and the printers John Higman and Wolfgang Hopilius.

[Pg 101] Venice, no date, but apparently before fifteen-hundred. ‘Dabaco che insegna a fare ogni ragione mercadantile: et a pertegare le terre con larte di la geometria: et altre nobilissime ragione straordinarie con la Tariffa come respondeno li pesi et monede de molte terre del mondo con la inclita citta de Vinegia. El qual Libbro se chiama Thesauro universale.’ Octavo.

The order of the operations is as in P. Borgi. The book is very full of florid ornaments and woodcuts.

Among books of the fifteenth century which I have had no opportunity of seeing are the following, described by Hain and others:

Without date. ‘Arithmeticæ Textus communis, qui pro Magisterio fere cunctis in gymnasiis ordinarie solet legi, correctus corroboratusque perlucida quadam ac prius non habita commentatione a Conrado Norico. Lipsiæ per Martinum Herbipolensem.’ Folio.

Without date, quarto, with the mark of Martin of Wurtzburg (Herbipolis). ‘Algorithmus linealis,’ beginning ‘Ad evitandum multiplices Mercatorum errores &c.’

Treviso, fourteen-seventy-eight, quarto. ‘L’arte del Abbacho, Practica molto bona et utile a chiachaduno qui vuole uxare l’arte merchandantia.’

Watt mentions an edition, Rome, fourteen-eighty-two, quarto, of the ‘Ludus Arithmomachiæ,’ attributed to John (he should have said William) Shirewood, bishop of Durham. Hain does not mention this.

Babenberg, fourteen-eighty-three, duodecimo. ‘Rechnungsbüchlein.’

Cologne, fourteen-eighty-five, quarto. ‘Ars numerandi, seu compendiosus tractatus de dictionibus numerabilibus.’

Strasburg, fourteen-eighty-eight, quarto. Anianus. ‘Compotus manualis magistri aniani. metricus cum commento Et algorismus.’

Leipzig, fourteen-eighty-nine, octavo. ‘Rechnung auf alle Kaufmannschaft.’ Murhard and Kloss (who cites Panzer) give this book to Joh. Widman, whose name appears in an address to the reader; but Hain, though naming Widman, puts the work down as anonymous.

Two collections of the works of Boethius, with the Arithmetic among them, Venice, fourteen-ninety-one and ninety-seven, folio.

Deventer (Daventrie), fourteen-ninety-nine, quarto. ‘Enchiridiom Algorismi sive tractatus de numeris integris.’

Without date, but with a letter from Balth. Licht dated fifteen-hundred, Leipzig, printed by Melchiar Lotter, quarto. ‘Algorithmus linealis cum pulchris conditionibus Regule detri: septem fractionum: regulis socialibus,’ &c. And another, seemingly of the same date, by the same printer, ‘Algoritmus linealis.’

No doubt the first English print on Arithmetic is cap. x. of ‘The Mirrour of the World or Thymage of the same,’ headed ‘And after of Arsmetrike and whereof it proceedeth,’ printed by Caxton in fourteen-eighty. (Peacock, p. 419, and Ames.)

[Pg 102] Basle, fifteen-three. Nicolas de Orbelli. ‘Cursus librorum philosophie naturalis venerabilis magistri Nicolai de Orbelli ordinis minorum secundum viam doctoris subtilis Scoti.’ Quarto in fours.

Florence (?), no date. Francisco di Lionardo Ghaligaio. ‘Summa De Arithmetica.’ Quarto in threes (N).

Florence, fifteen-sixty-two. Francesco di Lionardo Ghaligaio. ‘Pratica D’Arithmetica ... Nuovamente Rivista.’ Quarto in fours.

Two editions of the same book. The first must have been published before fifteen-twenty-three, the year in which Julius de Medicis, to whom it is dedicated as Cardinal, would have been [Pg 103] addressed as Pope. It is a complete and advanced treatise of arithmetic, in thirteen books. The tenth book begins algebra, or Arcibra, the introduction of which from the Arabic seems to be attributed to Guglielmo de Lunis; Leonard of Pisa and Giovanni del Sodo are also mentioned as writers.

Venice, fifteen-twenty-six. Joh. Fr. Dal Sole. ‘Libretto di Abaco novamente Stampato: Composto per lo excellente maestro Joanne Francisco dal sole Ingegnero: utilissimo a cadauno per imparare per se stesso senza maestro. Dimandato breve introductione.’ Octavo in twos (O).

A very elementary work, from the school of Borgi and Pacioli, proceeding up to the square root.

——— fifteen-forty-five. Gaspar de’ Texeda. ‘Suma De Arithmetica pratica y de’ todas Mercaderias Con la 5 orden de’ contadores. Hecho por Gaspar de’ Texeda, Con Privilegio Imperial.’ Quarto in fours (ON).

The colophon, which probably contains the place, is torn out. In numeration the places are distinguished throughout the first part of the book. Thus, 950777200000 is 950║777q⁰200║000, where ║ (the two lines should be joined at the bottom) stands for mill, and q⁰ for cuento, a million. The proof by sevens and nines appears, as in Borgi. The work is rather more advanced than that of the latter, and proceeds to the extraction of roots. The greater part is commercial. Rules for fractions are given, not complete. The rule of three opens with an invocation of the Trinity: Sfortunati goes further, for he connects the rule with the doctrine.

London, fifteen-forty-six. ‘An introduction for to lerne to recken with the pen, or with the counters accordyng to the trewe cast of Algorisme, in hole numbers or in broken newly corrected. And certayne notable and goodly rules of false positions thereunto added, not before sene in oure Englyshe tonge by the which all maner of difficile questions may easely be dissolved and assoyled Anno. 1546.’ Octavo. At the end is: ‘Imprynted at London in Aldersgate Strete by Jhon Herford.’

London, fifteen-seventy-four (printed by John Awdeley). ‘An introduction of Algorisme, to learn to recken wyth the Pen or wyth the Counters, in whole numbers or in broken. Newly overseen and corrected....’ Octavo (small) (O).

Rome, fifteen-eighty-three. Christ. Clavius. ‘Epitome Arithmeticæ Practicæ.’ Octavo (small) (O). See page 33.

Witeberg, sixteen-four. Gemma Frisius. ‘Arithmeticæ Practicæ Methodus Facilis.’ Octavo (small). See page 25.

See page 26. Macpherson (Annals of Commerce, eighteen-five, quarto, vol. i. p. 146) gives (from Anderson, i. 409) a French edition of Stevinus on book-keeping, Leyden, sixteen-two, folio, as quoted by Anderson from a copy in his own possession. The title he gives is ‘Livre de compte de prince à la manière d’ltalie en domaine et finance ordinaire: contenant ce en quoi s‘exerce le très illustre et très excellent Prince et Seigneur Maurice Prince d’Orange, &c.’

Frankfort, sixteen-thirteen. Joh. Faulhaber. ‘Ansa inauditæ et mirabilis novæ artis....’ Quarto. Same place, author, and form, sixteen-fourteen. ‘Numerus figuratus ...’ and sixteen-fifteen, ‘Mysterium Arithmeticum....’

These singular medleys of arithmetic, algebra, prophecy, and nonsense, seem to be all one publication. The date of the first is JVDICIVM, written at the bottom of the title-page. Alter the order of the letters into M.DC.VVJII., and it becomes 1613.

London, sixteen-fourteen. James Dowson. ‘De Numerorum Figuratorum Resolutione.’ Octavo.

This work gives Pascal’s table.

No place nor date. Joh. Harpur. ‘The Jewell of Arithmetick: or, the explanation of a new invented arithmeticall Table....’ Quarto.

London, sixteen-thirty-three. Joh. Johnson (Survaighor). ‘Johnson’s Arithmatick In 2 Bookes. The first, of vulgare Arithma: with [Pg 105] divers Briefe and Easye rules: to worke all the first 4. partes of Arithmatick in whole numbers and fractions by the Author newly Invented. The Second, of Decimall Arithmatick wherby all fractionall operations are wrought, in whole Numbers, in Marchants accomptes without reduction; with Interest, and Annuityes.’ Second edition. Duodecimo (O).

London, sixteen-forty-nine. R. B. (Mr of Arts). ‘Arithmetick Symbolical In one Book. In which the Mystery of Numeration by Symbols is revealed.’ Octavo.

A short and easy treatise on Algebra.

London, sixteen-fifty-five. R. B. ‘An Idea of Arithmetick at first Designed for the use of the Free-Schoole at Thurlow in Suffolk. By R. B. Schoolmaster there.’ Octavo (O).

Decimals and algebra, all on the plan of Oughtred.

Antwerp, sixteen-sixty-two. Jean Raeymaker. ‘Traicte d’Arithmetique Contenant les quatres especes, avec la regle de trois, et la Practique.’ Octavo.

Not a first edition. Nothing but questions and their answers, as in John Speidell’s work of sixteen-twenty-eight.

London, sixteen-seventy-five. C. H. ‘The Golden Rule made Plain and Easie.’ Octavo. Second edition.

No date nor place (I think about sixteen-eighty). J. W. (of Brandon). ‘A progression in Arithmetical Progression ... with Some things very remarkable in that Mystical Number 666.’ Octavo in twos.

The author shews how to form squares and cubes by differences. He is very charitable, for his day, to the Papists: for, setting down twelve as the divine number, and that of the apostles, he seems to imply that the Roman church, as having a number made up of sixes, may be half Christian and half apostolic.

London, sixteen-eighty-four. —— ‘Enneades Arithmeticæ; the Numbering Nines. Or, Pythagoras His Table extended to All Whole Numbers under 10000.’ Octavo in twos.

The plan is, to have 99 rods instead of nine, in Napier’s set; so as to have the first nine multiples of all numbers short of 100.

[Pg 106] London, sixteen-eighty-five. John Wallis. ‘A treatise of Algebra, both Historical and Practical.’ Folio in twos.

London, sixteen-ninety-three. John Wallis. ‘De Algebra Tractatus; Historicus et Practicus. Anno 1685 Anglice editus; Nunc Auctus Latine.’ Folio in twos.

The Latin, or augmented edition, is the second volume of Wallis’s collected works. See the Introduction to this Catalogue.

London, sixteen-ninety-seven ——. ‘Computatio Universalis, seu Logica Rerum. Being an Essay attempting in a Geometrical Method, to Demonstrate an Universal Standard, whereby one may judge of the true Value of every thing in the World, relatively to the Person.’ Octavo.

The author, cutting down life to its half, to allow for childhood, sickness, &c., proceeds to a calculation how the true value of every thing is to be estimated numerically. This, and Craig’s more famous book (see Useful Knowledge Library, Probability,⁠[21]) are consequences of the Principia. It is curious that as soon as force was widely made known as an object of numerical measurement, some people began to fancy prudence, pleasure, and pain could be submitted to the same process.

Breslau, seventeen-seventy-nine. Joh. Ephr. Scheibel. ‘Einleitung zur mathematischen Bücherkentnis. Eilftes Stück.’ Octavo (small).

See the Introduction. This part contains the arithmetical bibliography; but the paging does not begin after that of the ‘Zehntes Stück.’ In fact, the bibliography of this book of bibliography is a study of itself.

Göttingen, seventeen-ninety-six, ninety-seven, ninety-nine, eighteen-hundred. Abr. Gotthelf Kastner. ‘Geschichte der Mathematik....’ Four volumes octavo.

See the Introduction, as to Kästner.

OF REPORTED AUTHORS, EDITORS, &c. OF WORKS ON ARITHMETIC,
INCLUDING THE INDEX TO THIS CATALOGUE.

The numbers refer to pages of the present work; and the (‡) prefixed shews that
the author has been referred to by Dr. Peacock in his History of Arithmetic.

• J. H. M. A., 68.
• Abercromby, 51.
• Abraham, Aben Ezra, 95.
• Abraham, Chaja.
• Abraham of Prague.
• Abt.
• Adami.
• Adams.
• Addington.
• ‡Adelbold.
• Aeneæ.
• Agrippa, 13, 21.
• Ahlwardt.
• Albert.
• Alberti.
• Aldhelm.
• Alexander de Villa Dei.
• Alfraganus.
• ‡Algus, 92.
• Alkindus.
• Allingham.
• ‡Alsted, 41, 42.
• Ammonius, Joach.
• Anatolius, 88.
• Anderson, 76, 93.
• Andreas.
• ‡Andrews, Laur.
• Anianus, xvi. 101.
• Anjema.
• Ankelin.
• Apian.
• ‡Apollonius, 76.
• Apuleius, 4.
• Archimedes, 76, 84.
• Archytas.
• Argyrus.
• Aristarchus.
• Arnold.
• Artabasda (Nic. Smyrn.), 34.
• Aryabhatta.
• Asclepius.
• Ashby.
• Astius, 88.
• Atkinson, 65.
• Atwood, 77.
• Aub.
• Augustine.
• Ault, 98.
• Author.
• D’Autrêpe.
• Avemann.
• Aventinus.
• Awdeley, 103.
• Aylmer.
• Ayres, 64.
• R. B., 105.
• ‡Bachet de Mezeriac, 47.
• Bachmeister.
• ‡Bacon, xix. 14.
• Bagley, 97.
• Baily, 63.
• ‡Baker, 23, 25, 50.
• Barlaam, 32.
• Barlow.
• Barnes. [Pg 108]
• Barozzi.
• Barreme, 75.
• Barres, des.
• Bartel.
• Bartjen, 48.
• Bartiens.
• Barth.
• Bartl.
• ‡Bartschius.
• Bartoli.
• Barton.
• Bartrum, 98.
• Barwasser.
• Basedow.
• Baselli, 77.
• Bassi.
• Baum.
• Bausser.
• Bayer.
• Bayley.
• Beasley.
• Beausardus.
• Becker.
• Beckman.
• ‡Beda, 34.
• Bedwell, 35.
• Behen.
• Behm.
• Behmen.
• Behrens.
• Bellie.
• Benedictus, J. B., 31.
• Benese, 18.
• Berckenkamp.
• Berger.
• Berghaus.
• Bergsträste.
• Berkeley, 63.
• Bernard.
• Bernoulli, 78.
• Berthevin.
• Bertram.
• Bettesworth, 71.
• Beutel.
• Beuther.
• ‡Beverege, 66.
• Bevern.
• Beyenberg.
• Beyern.
• Beyerus.
• Bezout, 87.
• ‡Bhascara Acharya, 86.
• Bidder, 94.
• Biagio da Parma.
• Biermann.
• Bierögel.
• Bildner.
• Biler.
• Billy, De.
• Billingsley.
• Binet.
• Birkin, 80.
• Birkner.
• Birks, 75.
• Bischof.
• Blagrave, 25.
• Blasing.
• Blasius.
• Blassière.
• Blavier, 75.
• Bluhme.
• Blundevile, 9, 30.
• Böbert.
• Bockel.
• Bocmann.
• Bode.
• Boden.
• Boeclerus.
• ‡Boethius, xx, 3, 4, 5, 10, 11, 13, 17, 100, 101.
• Böhm.
• Boissière.
• ‡Bombelli.
• ‡Bonacci (Leonard of Pisa).
• Boninus.
• Bonneau.
• Bonnycastle, 76, 91.
• ‡Borgi, Piero, 2, 98.
• Borgo, Pietro, 2.
• ‡Borgo, Di (Paciolus), 2.
• Boscherus.
• Boteler, 44.
• Bourdon, 91.
• Bouvelin.
• Bovillus, 3.
• Boyer.
• Boysen.
• Bradwardine, 11.
• ‡Bragadini, 11. [Pg 109]
• ‡Brahmegupta, 86.
• Brancker.
• Brand, 63.
• Brandauen.
• Brander.
• Brandt.
• Brasser, 36.
• Bredon.
• Brent.
• Bretschneider.
• Brett.
• ‡Bridges, 44, 60, 65.
• ‡Briggs, xxv. 42.
• Brodhagen.
• Brodreich.
• Bronchorst.
• Broschius, 43.
• Brown, 62, 65, 69, 84.
• Bruell.
• Bruncke.
• Brunetti.
• Bruno (Jordano), 28.
• Brunot.
• Brunus.
• Bubenkius.
• Büchner.
• ‡Buckley, vii. xvi. 20.
• Budæus, 20.
• Bullialdus, 41.
• ‡Bungus, 28.
• Bunsen.
• Bunzel.
• Burckhardt.
• Burtius.
• Buscher, 28.
• Bussche.
• Busse.
• Bütemeister.
• ‡Buteo, 22.
• Butler, 38, 87.
• Butte.
• Butterman, 97.
• Buttner.
• Buxerius.
• Cadell.
• Cæsar (Jul. Patav.).
• Caius.
• Calandri, 1.
• Caligarius (Pelacanus).
• Calmet.
• Camerarius, 46.
• Campanella.
• ‡Canacci.
• Cap de Ville.
• ‡Capella (Martianus).
• Cappaus.
• Capricornus.
• Caraldus (?), 33.
• Cardan, 17.
• Cardinael.
• Carey, 88.
• Carlile, 79.
• Carolus.
• Carr.
• Carver, 87.
• Cassini, D.
• ‡Cassiodorus.
• Castiglione, 73.
• Cataldus, xxi. 33, 35.
• ‡Cataneo.
• Cathalan.
• ‡Caxton, 101.
• Ceulen.
• Chalosse.
• Chamberlain.
• Chambers, 32, 98.
• Champier.
• Champion, 67.
• Chapelle.
• Chapman.
• Chappell, 80.
• Charlton, 91.
• Chauvet.
• Chauvin.
• Chelius.
• Chernac.
• Chladenius.
• Chlichtoveus, 3.
• ‡Crishna.
• Christiani.
• Christofle.
• Chuno.
• Cirvello.
• Claircombe.
• Clapproth.
• Clare. [Pg 110]
• Clark, 49, 85.
• Clarke, 79.
• Classen.
• Clausberg.
• Clavel.
• ‡Clavius, 7, 33, 76, 104.
• Clay, 36.
• Clebauer.
• Cleomedes.
• Clerk.
• Clermont.
• Cloot.
• ‡Coburgk (Sim. Jac.).
• Cock.
• ‡Cocker, xxii. 56, 65, 92.
• Cockin, 74.
• Coetsius, H.
• Coggeshall, 52, 65.
• Coignet.
• Colburn, 94.
• Cole.
• ‡Colebrooke, 86.
• Coles.
• Collins, 48, 50, 51, 57.
• Colonius.
• Colson.
• Condillac, 80.
• ‡Condorcet, 53, 82, 86.
• Copeland.
• ‡Cornaro.
• Cortes.
• Cossali.
• Cotes.
• Counteneel.
• Coutereels.
• Cracher.
• Crelle.
• Creutzberger.
• Crivellius.
• Crohn.
• Crosby, 80.
• Crousaz.
• Crowquill, 96.
• Cruger.
• Crumpton, 97.
• Crusius.
• Cuetius.
• Cunn, 64, 69, 74.
• Cunningham, 98.
• Cuno.
• Cuppaus.
• Curtius.
• Cusa, 10.
• Cusanus, Joh., 10.
• Daetrius.
• Dafforne, 50.
• ‡Dagomari.
• Van Damme.
• Dangicourt.
• Dansie.
• Danxt.
• Danziger.
• Daries.
• Darnell, 90.
• Dary, 48.
• ‡Dasypodius.
• Dauden.
• Daumann.
• Davenant.
• Daviden.
• Davidson, 71, 98.
• Davies, 19, 30, 89, 95.
• Davis, 86.
• Dean, 97.
• Dechales, xv. 53.
• Decker.
• Dedier.
• ‡Dee, 22.
• Degen.
• Degrange, 97.
• Deidier.
• Deighan, 83.
• ‡Delambre, xvii. 84.
• Delile, 97.
• De Morgan, 89, 96.
• Detri.
• Deubelius.
• Develay.
• Develey, 97.
• Dibuadius, 32.
• Dicellus.
• Diezer.
• Digges, 24, 28, 40.
• Dilworth, 75, 76. [Pg 111]
• ‡Diophantus, 46.
• Ditton, 66, 70.
• Dluskin.
• Dodson, 71, 73.
• Döhren.
• Donn, 72.
• Döppler.
• Dowling, 74, 98.
• Dowson, 104.
• Drape.
• Dubost, 83.
• Dühren.
• Duke.
• Duncombe, 78.
• Dunkel.
• Dunus.
• Dupp.
• Düsing.
• Dyer, 97.
• Eadon, 97.
• Eberenz.
• Eggerer.
• Ehrhardi.
• Eichstadius.
• Elend.
• Elf.
• Elias Misrachi.
• Emerson, 78.
• Enderlin.
• Engelbert.
• Engelbrecht.
• Engelken.
• English.
• Eratosthenes.
• Erhorn.
• Erxleben.
• Eschenberg.
• Eslingen.
• ‡Euclid, 63, 77.
• Euler, 78.
• Euteneuer.
• ‡Eutocius, 84.
• Everard, 68.
• Ewing.
• Faber (Stapulensis), 3, 10.
• Falconius.
• Famuel.
• Faulhaber, 104.
• Farrar, 87.
• Favre.
• Fayst.
• Feist.
• ‡Feliciano da Lazesio.
• Felicianus.
• Felkel.
• Felton, 98.
• Fenning, 71.
• Fermat, 47.
• Fernel, 8, 15.
• Ferrei.
• ‡Fibonacci (Leonard of Pisa).
• Finckius.
• ‡Fineus (Orontius), viii. 5, 15, 16, 18, 19.
• Fischer.
• Fisher, 34.
• Fissfeld.
• Fletcher.
• Flicker.
• Flor.
• Fludd, 35.
• Flügel.
• Follinus.
• Fontaine.
• Forbes, 48.
• Forcadel.
• Forir, 96.
• Fortia, 79.
• Fortius, 16.
• Forster, 38.
• Foster.
• Foys de Vallois.
• Fraiter, 97.
• François.
• Fraser, 80.
• Freber.
• ‡Freigius.
• Frend, 84.
• Frenicle, 53.
• Frey.
• Friedeborn.
• Friesenborch.
• Friess.
• ‡Frisius (Gemma), 25, 104. [Pg 112]
• Frömmichen.
• Frommius.
• Fuhrmann.
• Fuller.
• Funk.
• Funcke.
• Furit.
• Fürderer.
• Furtenbach.
• Fustel.
• Gadesby, 97.
• Gaesse.
• Galigai (Caligarius).
• Gallimard.
• Galstrupius.
• Galtruchius, 50.
• Galucci, 5.
• ‡Ganesa.
• Garrett, 91.
• ‡Garth.
• Gartner.
• Gaudlitz.
• Gauss.
• Gautier.
• ‡Geber.
• Gebhardus.
• Gehrl.
• Geigers.
• Gelder, De, 41.
• ‡Gemma Frisius, 25,104.
• Gemma Reinerus.
• Gempelius.
• Gendre, Le.
• Gerardus.
• Gerardus (Paul).
• ‡Gerasenus (Nicomachus).
• ‡Gerbert.
• Gerbier, 51.
• Gerhardt.
• Gerhard.
• Gernstein.
• Gersten.
• Geyger.
• Ghaligaio,}
• Ghaligai,  } 102.
• Gibson, 44.
• Gietermaker.
• Giordano.
• ‡Girard, Alb. 26, 37.
• Girtanner, 88.
• Girthanner.
• Glareanus (Loritus).
• Gleich.
• Gleitsmann.
• Glogovia, Joh. de.
• Glynzonius.
• Gmunden, 11.
• Goffen.
• Goldammer.
• Goldberg.
• Gollner.
• Good, 65.
• Goodacre, 83.
• Goodwyn.
• Goosen.
• Gordon, 97.
• Gore.
• ‡Gosselin, 21.
• Gottignies, 51.
• ‡Gottlieb.
• Göttsche.
• Göttschling.
• Gough, 79.
• Graf.
• Grafe.
• Graff, De.
• Gräffe.
• Graffinriedt.
• Grammateus.
• Grandi.
• Graumann.
• Grave.
• Gravesande, S’, 73.
• Gray, 89,95.
• Gregory, 85.
• Greig.
• Gretser.
• Greve.
• Grey, 90.
• Griesser.
• Grillet.
• Grossi.
• ‡Grostete. [Pg 113]
• Gruneberg.
• Grünewald.
• Grüning.
• Grünmos.
• Grupe.
• Gruson.
• Grysen.
• Gryson.
• Guden.
• Gueinzius.
• Gujieno (Siliceus), 15.
• Gunter, xxv.
• Gunther.
• Gütle.
• Guy.
• C. H., 105.
• J. H., 68.
• Haas.
• Habelin.
• Hagen.
• Hager.
• Haidinger.
• Haintzelmann.
• Halbert.
• Halfpenny, 70.
• ‡Halifax (Sacrobosco).
• ‡Halley, 37, 64, 67.
• Halliday.
• Halliman, 52, 69.
• Halliwell, 14, 15, 30.
• Hämel.
• Hamelius (Paschasius).
• Hamilton.
• Hänel.
• Hardcastle.
• Hardy, 76.
• Harpur, 104.
• Harriot, 42.
• Harris.
• Harrison, 94.
• Harroy.
• Hartmann.
• Harttrodt.
• Hartwell, 22.
• Hartwich.
• Hase.
• Hasius.
• Hatton, 22, 66, 67.
• Hauff.
• Haughton, 38.
• Haupt.
• Hauser.
• ‡Hawkins, 52, 56, 58, 59, 65.
• Hawney, 65.
• Hay, 97.
• Hayes.
• Heath, 71.
• Heber.
• Heckenberg.
• Hederich.
• Hedley.
• Heere.
• Heilbronner, xv. 69.
• Hein.
• Heinatz.
• Heinlin, 54.
• Heinsius.
• Heinzelmann.
• Held.
• Helden.
• Hell.
• Hellwig.
• Helmreich.
• Hemeling.
• ‡Henischius, 33.
• Hennert.
• Henning.
• Henrici.
• Hentsch.
• Hentschel.
• Hentz.
• Hentzen.
• Herberger.
• Herbestus.
• Herford, 103.
• Herigone, 40.
• Hermann.
• Hermantes.
• Herodian.
• Herr.
• Herrmann.
• ‡Hervas.
• Herwart.
• Hesse. [Pg 114]
• Heyen.
• Heyer.
• Heynatz.
• Hill, 70.
• Hills (Hylles?).
• Hindenburg.
• Hinrichsen.
• Hirsch.
• Hoar, 97.
• Hobelius.
• Hochstetter.
• Hodder, 46, 57, 58, 59, 60.
• Hodges.
• Hodgkin, 67.
• Hoff.
• Hofflin.
• Hoffmann.
• Hohenberg (Herwart).
• Hohnstein.
• Holiday.
• Holland.
• Hollenberg.
• Holmes.
• Holywood (Sacrobosco).
• Holzmann (Xylander).
• Holzwart.
• Hood, 24.
• Hoorbeck.
• Hope, 78.
• Horem, 11.
• Horner, xxvii. 88, 89.
• ‡Hostus.
• Hotson, 98.
• Howell.
• Howes.
• Hoyer.
• Hübsch.
• Hudalrich (Regius).
• Huet.
• Hugerus.
• Hullie du Pont, L’.
• Hume, 10.
• Hunger.
• Hunichius.
• Hunt, 39.
• Hurry.
• Huswirt, 3.
• Huth.
• Hutton, 85, 89, 98.
• Hylles, 31.
• Hyperius.
• Ihne.
• Illing.
• Imison.
• Imhof.
• Imhoof.
• Impen.
• ‡Ingpen, 36.
• Ingram, 85, 88.
• Intelman.
• Ion.
• Irson.
• Isidorus.
• Jackson.
• ‡Jacob (Coburgk).
• Jacobi.
• Jacobs.
• Jagar.
• Jager.
• Jahn.
• ‡Jamblichus, 17, 46, 88.
• Jaquinot.
• Jarvis, 98.
• Jaster.
• ‡Jeake, 55.
• Jesper.
• ‡Joch.
• ‡Johnson, 59, 104.
• Johnston, 86.
• Joncourt.
• ‡Jones, 63, 69.
• Jons.
• Jordaine.
• Jordan.
• Jordanus, 10, 100.
• ‡Josephus Hispanus.
• Josseaume.
• Joyce, 90. [Pg 115]
• Jung.
• Juvigny, 97.
• Kaltenbrunner.
• Kampke.
• Kappell.
• Karrer.
• Karsten.
• ‡Kästner, xv. 106.
• Kate.
• Katen.
• Kaudler.
• Kauffunger.
• Kaukol.
• Kearsley.
• Keckermann.
• Keegan.
• Kegel.
• Keiser.
• Keith, 78, 95.
• Kellerman.
• Kempten, 23.
• ‡Kepler, S.
• ‡Kersey, 48, 58, 73.
• Killingworth.
• ‡King, 83.
• Kirby.
• ‡Kircher.
• Kirchner.
• Kirkby, 67, 71.
• Klinger.
• Klippstein.
• Klügel.
• Knoes.
• Knorre.
• Knott.
• Köbel, 9, 10, 11.
• Koch.
• Kochanskus.
• Köhler.
• Kolbe.
• Königsbrun.
• Königstein.
• Kopffer.
• Koster.
• Krafften.
• Kraft.
• Kramer.
• Krause.
• Kreitscheck.
• Kretschmar.
• Kritter.
• Kroymann.
• Krüger.
• Krünitz.
• Kruse.
• Kundler.
• Küpfer.
• Küster.
• Du Lac.
• Lacher.
• Lacroix, 87.
• Lagny, 55.
• ‡Lagrange, 82.
• Lambech.
• Lambert.
• Lamboy.
• Lamotte.
• Lamy, 53.
• Landase.
• Landus.
• Lange.
• Langens.
• Langius.
• Langner.
• Lantz.
• ‡Laplace, 82.
• Lardner.
• Latomus.
• Launay.
• Laurbech.
• ‡Lauremberg, 36, 41.
• Lauterborn.
• Lauxius, 10, 100.
• Lavus.
• Lawson, 49.
• Lax, 11.
• Leadbetter, 68, 70.
• Lebez. [Pg 116]
• Leblond.
• Lecchi.
• Lechner.
• Leeuwen, Van.
• ‡Leibnitz.
• Leichner.
• Leiste.
• Leistikov.
• Lembke.
• Lemoine.
• Lempe.
• Lenz.
• ‡Leonard of Pisa (Bonacci Fibonacci), xviii. 1, 17, 94, 95.
• Leotaud, 45.
• Leri.
• ‡Leslie, xvi. 89.
• ‡Leupoldus.
• Levera.
• Levi.
• Leybourn, 46, 49, 50, 52, 54, 72, 89.
• Libri, xvii. 93, 95.
• Licht, 101.
• Lichtenberg.
• Lidonne.
• Liebe.
• Lieberwirth.
• Liedbeck.
• Liesganig.
• Lindenbergius.
• Lindner.
• Lintzner.
• Lipstorp.
• Liset, 50.
• Little, 80.
• Littmann.
• Liverloz.
• Loche.
• Locher.
• Lohenstein.
• Loher.
• La Londe.
• Longomontanus.
• Lonicerus.
• Lorenz.
• Loritus (Glareanus).
• L’Ortie, }
• Lortse,  } 12, 102.
• Lory.
• Lösch.
• Loscher.
• Lossius.
• Lostau, 67.
• Lotter, 101.
• Lovell.
• Löw.
• Lowe, 70.
• Lucar, 28.
• Lucas.
• Lucas à St. Edmondo.
• Lucius.
• Luders.
• Ludolff.
• Ludwig.
• Lunenzius.
• Lunesclos.
• ‡Lunis, Gul. de, 102.
• Lunze.
• Luthardt.
• Luya.
• Lydal.
• ‡Lyte, 36.
• J. R. M. 62.
• Machin, 74.
• Maes.
• Magelsen.
• Mainwaring.
• ‡Mair, 56, 77.
• Malapertius.
• ‡Malcolm, 66, 69.
• Maler.
• Malham.
• Malleolus.
• Mandey, 54.
• Manning.
• ‡Manoranjana.
• Mantaureus.
• Maphæus.
• Marcellus.
• Marchetti.
• Mariani.
• Markham.
• Marleen, 97. [Pg 117]
• Marsh, 69.
• Marshall.
• Martin, 68, 69, 73.
• Martini.
• Martino.
• Martinus (Herbipolensis), 101.
• Maslard.
• Massard.
• Masterson, 29.
• Mather.
• Matheson, 97.
• Mauduit.
• Maurolycus, 24.
• May.
• Mayer.
• Maynard, 80, 83, 95.
• Mayne, 48.
• Mazeas.
• Mazzonius.
• Meanus.
• Medler.
• Meenenaer.
• Megillus.
• Meierus.
• Meinck.
• Meissner.
• ‡Mellis, 22, 27.
• Melrose, 88.
• Mengolus.
• Menher, 23, 33.
• Mercator, 98.
• Merckel.
• Mereau.
• Meres.
• Merliers.
• Mesa.
• Messen.
• Metius, 41.
• ‡Metrodorus.
• Metternich.
• Meurerus.
• Meursius.
• Meyer.
• Meynier, 34.
• Mezburg.
• ‡Mezeriac (Bachet).
• Michaelis.
• Michahelles.
• Michelsen.
• Micraelius.
• Micyllus.
• Middendorpius.
• Milns, 80.
• Minderlein.
• Mirus.
• Mittendorf.
• Mognon.
• ‡Mohammed ben Musa, 93.
• Mole.
• Molineux, 79.
• Möller.
• Monhemius.
• Monson.
• Monzon.
• ‡Moore, 43, 49, 52, 59.
• Morin.
• Morisotus.
• Morland, 47.
• Morrison, 97.
• Morsshemius.
• Morssianus.
• Moscopulus (Emanuel).
• Mose.
• Moser.
• Moses.
• Mosley, 98.
• Mülich.
• Müller.
• Mullerus.
• Müllner.
• Mullinghausen.
• Munnoz.
• Muris, J. de, 3, 4, 11.
• Nabod. }
• Naiboda. }
• Nagel.
• Napier (Neper).
• Nefe.
• Nelkenbrucher.
• Nemorarius (Jordanus).
• ‡Neper, xxiii. 35.
• Nesius.
• Neudorffer.
• Neufville. [Pg 118]
• Neumann.
• Newton, 46, 73.
• ‡Nicholas Smyrnæus Artabasda, 34.
• Nicholson.
• Nicolson, 97.
• ‡Nicomachus (Gerasenus), 17, 46, 88.
• Niese.
• Nisbet.
• Nomer.
• Nonancourt.
• Nonius (Nunez), 23.
• Nottnagel.
• Norfolk, 15.
• Norry.
• ‡Norton, 27.
• Noviomagus, 18, 34.
• Nunez, 23.
• Obereit.
• Ochler.
• Odo.
• Oelse.
• Oeser.
• Oesfeld.
• Oesterle.
• Ofenlach.
• Ohm.
• Olaus.
• ‡Oldcastle, 28.
• Opuntius.
• Orbelli, 102.
• ‡Orontius (Fineus), viii. 5, 15, 16, 18, 19.
• ‡Ortega, 102.
• L’Ortie, J. de, 102.
• Oser.
• Ostermann.
• Ostertag.
• Osterwald.
• Otto.
• ‡Oughtred, xiv. xxii. xxv. 37, 38, 49.
• Ouvrard.
• Ouvrier de L’Isle.
• Overheyden.
• Ozanam.
• W. P., 30.
• Pabst.
• Pachymeres, Geo.
• ‡Paciolus (di Borgo), xx. 2, 95.
• ‡Pajottus.
• Palmquist.
• ‡Paolo dell’ Abacco.
• ‡Pappus, 85.
• Pardon, 68, 69.
• Parent.
• Parker, 93.
• Parisius.
• Parrot.
• Parsons, 63.
• Partridge, 42, 51.
• Pascal.
• Paschasius (Hamelius).
• Pasi.
• Paterson.
• Patrick.
• Pauer.
• Paulinus.
• Paulinus à St. Josepho.
• Paulus.
• Paze.
• Peacock, xvii. xxiv. 14, 91, 97.
• ‡Peele, 28.
• Pelacanus.
• ‡Peletarius, }
• Pelletier, } 22, 25.
• Pelicanus.
• Pell.
• Penkethman, 41.
• Penrose, 97.
• Da Perego.
• Perez.
• ‡Pergola.
• Perry, 98.
• Pesare, Leon. de.
• Pescheck.
• Petersen.
• Petreius. [Pg 119]
• Petri (Nicol.), 31.
• Petvin.
• Petzoldt.
• Peucerus.
• Peurbach, 11.
• Peverone, 21.
• Peyrard, 84, 86.
• Pfaff.
• Pfeffer.
• Pfeiffer.
• Pflugbeil.
• Philipp.
• Philippus Opuntius.
• Philips.
• Phillips, 97.
• ‡Philo.
• Philoponus.
• ‡Photius.
• Pickering.
• Pierantonio.
• Pieterson.
• Pike.
• Pisani.
• Piscator.
• Planer.
• ‡Planudes.
• Platin.
• Platz.
• Playfair, 81.
• Playford, 72.
• Plotinus.
• Poeppingius.
• Poetius.
• Polenus.
• Poppin.
• ‡Porphyry.
• Porter.
• Pöschmann.
• Postellus.
• Potter.
• Pratt, 104.
• Prestet, 54.
• Preston, 88.
• Prexendorffer.
• Printz.
• Pritchard, 90.
• ‡Proclus.
• Prosdocimo di Beldomando.
• Psellus, 21.
• ‡Ptolemy, 84.
• Purdon, 87.
• Purser.
• Putsey, 98.
• Pütter.
• ‡Pythagoras, 1, 4.
• Quensen.
• Rabus.
• Rademann.
• Raeymaker, 105.
• Ram.
• Ramsbottom, 97.
• ‡Ramus, 29.
• Randall, 74.
• Ranzovius.
• Raphson, 66, 74.
• Rawlin.
• Rawlyns.
• Raymaker.
• Rea, 65.
• Recorde, xxi. 21, 22, 59, 66.
• Rees.
• ‡Regiomontanus, xiv. 16, 91.
• Regius, Hudalrich, 16.
• Regneau.
• Rehmann.
• Reich.
• Reiche.
• Reichel.
• Reimann.
• Reimer.
• Reinau.
• Reiner.
• Reinhard.
• Reinhold.
• Reinholdt.
• Reisch, 4.
• Reiser.
• Remer. [Pg 120]
• Remmelinus.
• Renaldini.
• Renaldus.
• Renterghem.
• Resenius.
• Ressler.
• Reuss.
• Reyher.
• Reymer.
• Reynaud, 87.
• Reyneau.
• Reynolds, 97.
• Rhaetus, 15.
• Rheticus, xxi.
• Rho.
• Riccius.
• Richards, 97.
• Richardson.
• Richter.
• Riederus.
• Riege.
• Riese, 19.
• Rigaud, 19, 76.
• Riley.
• Ringelbergius, 16.
• Ritchie, 20, 88.
• Rivail, 97.
• Rivard.
• Rivet, 74.
• Robertson.
• La Roche (Villefranche), 12.
• Roder.
• Romer.
• Romanus.}
• Roomen, Van. }
• Rommel.
• Ronner.
• Roscher.
• Roselen.
• Rosen, 93.
• Rosenkranz.
• Rosenthal.
• Rosenzweig.
• Roslin.
• Rossel.
• Roth.
• Rouquette.
• Le Roux.
• Rouyer.
• Röver.
• Royer.
• Rozensweig.
• Ruchetta.
• Ruddiman, 68.
• ‡Rudolff, 23, 104.
• Ruffus, 13.
• Rühl.
• Ruhmbaum.
• Ruinellus.
• Russell.
• Rust.
• Rutherford, 95.
• Ryff, 9.
• R. S. S., 62.
• ‡Sacrobosco (Holywood, Halifax), xix. 13.
• Sadler, 75.
• Salfeld.
• ‡Salignac.
• Salmasius.
• Salomon, 75.
• ‡Sanchez.
• Sarret.
• Sauer.
• Saul.
• ‡Saunderson, 68, 71.
• Savary, 98.
• Savonne.
• Saxon.
• Scalichius.
• Schäffer.
• Scharf.
• Schatzberg.
• Schedel.
• Scheffelt.
• Scheibel, xv. 106.
• Schellenberg.
• Schelliesnig.
• Schessler.
• Scheubelius, 20.
• Schey.
• Schiener.
• Schiller.
• Schirmer. [Pg 121]
• Schlee.
• Schlegel.
• Schleupner.
• Schlönbach.
• Schlosser.
• Schlügel.
• Schlüssel.
• Schmalzried.
• Schmid.
• Schmidt.
• Schmotther.
• Schneidt.
• Schonberger.
• Schoner, 16, 29, 35.
• Schoop.
• Schöttel.
• ‡Schott, 45.
• Schramm.
• Schreckenberger.
• Schreckenfuchsius.
• Schreyber.
• Schreyer.
• Schröter.
• Schübler.
• Schuler.
• Schulze.
• Schultze.
• Schumacher.
• Schürmann.
• Schurz.
• Schwarzer.
• Schweighäuser.
• Scoten, Van.
• Scott, 97.
• Seck.
• Seckgerwitz.
• Segner.
• Segura.
• Selden.
• Seller.
• Sempilius.
• Seriander.
• Servin.
• ‡Severinus, Boethius.
• ‡Sfortunati, 16, 103.
• Shakerley.
• Sharpe.
• Shelley, 73.
• Shepherd.
• Shield.
• Shirewood, 10, 101.
• Shirtcliffe, 69.
• Siegel.
• Silberschlag.
• Siliceus, 15.
• Simpson, 71.
• Sinclair.
• Sinner.
• Siverius.
• Smart, 63.
• Smeeth, 97.
• Smith, 72.
• ‡Smyrnæus (Nich. Artabasda), 34.
• Smyters.
• Snell, 64.
• ‡Snellius, 32.
• Soave.
• Soc. Usef. Kn. 93.
• Soc. Chr. Kn. 98.
• Sole, Dal, 103.
• Sosen, Von.
• Sotter.
• Speidell, 37.
• Speusippus.
• Spies.
• Spiess.
• Spitzer.
• Splittegarb.
• ‡Srid’hara.
• Stadius.
• Staps.
• Stapulensis (Faber).
• Starcken.
• Stark.
• Starke.
• Steger.
• Steinius, 25.
• Steinmetz.
• Stenius.
• Stephanus à St. Gregorio.
• Stephens.
• Sterk, 16.
• Stern.
• ‡Stevinus, xxiii. 26, 32, 35, 104.
• Steyn.
• Sthenius.
• ‡Stifelius, 19, 23.
• Stigelius.
• Stillinger. [Pg 122]
• Stimmingen.
• Stock.
• Stöffler, 8.
• Stonehouse, 72.
• Storr.
• ‡Strachey, 86.
• Strauchius.
• Stricker.
• Strigelius.
• Stritter.
• Strozzi.
• Strubius.
• Strunze.
• Sturmius, 62.
• Suevius (Sigism.).
• Suisset, 12.
• Sulzbach.
• Sutton, 76.
• Svanberg.
• Tabingius.
• Tabouriech.
• Taccius.
• ‡Tacquet, 56, 63.
• Tacuinus, 34.
• Taf.
• Tait.
• Tallen.
• Tangermann.
• Tannstetter, 11.
• Tap, 33.
• Tarragon.
• ‡Tartaglia, 21.
• Tassius.
• Tate, 85, 96.
• ‡Taylor, 76, 86, 87, 90, 97, 98.
• Tedenat.
• Telauges.
• Telfair.
• Tennulius, 46.
• ‡Teruelo.
• Tessaneck.
• Tetens.
• Texeda, 103.
• ‡Theon, 41, 84.
• Theophrastus.
• Thevenau.
• Thierfelders.
• Thoman.
• Thompson, 80, 98.
• Thornycroft.
• Thorpe, 97.
• Thoss.
• ‡Thymaridas.
• Timaus.
• Tinwell, 91.
• Tissandeau.
• Tocklerus.
• Toissonière.
• Tollen.
• ‡Tonstall, xv. 13.
• Torchillus (Morssianus).
• Trapp.
• Treiber.
• Trenchant.
• Treu.
• Treyrerens.
• Trincano.
• Trommsdorf.
• Trotter, 80, 85.
• Turner.
• Tylkowski.
• Tyson, 91.
• Tzechani.
• Udalrich (Regius).
• Ulman.
• Ursus.
• Ursinus.
• ‡Urstisius, 24.
• J. B. V., 84.
• Valla, 3.
• Vallerius.
• Vandamme. [Pg 123]
• Vandenbussche, 23.
• Vandendyck.
• Vandervelde.
• ‡Vasara.
• Vayra.
• Vega, 89.
• Veiar.
• Velhagen.
• Vellnagel.
• Veronensis.
• Vicar.
• Vicecomes (Visconti), 24.
• Vicum.
• Vierthaler.
• Vieta, xxi. 42.
• Villefranche.
• Vincent.
• Vinci (Leonardo da), 20.
• Vinetus.
• ‡Virgilius Salzburgensis.
• Visconti, 24.
• Vitalis.
• ‡Vlacq.
• Voch.
• Vogel.
• Voigt.
• Volck.
• Vollimhaus.
• Vomelius.
• Voster.
• Vries.
• Vulpius.
• Vyse, 81.
• J. W., 43, 105.
• Wagner.
• Walbaum.
• Walgrave.
• Walker, 86, 90.
• Walkingame, 80, 96.
• ‡Wallis, xiv. xxii. 37, 38, 44, 69, 82, 85, 106.
• Walrond.
• Walther.
• Waninghen.
• Warburton, 81.
• ‡Ward, 65, 69.
• Waserus.
• Wastell, 63.
• Watson, 98.
• Weberus.
• Webster, 40, 68.
• Wechselbuch.
• Wecke.
• Weddle, 89.
• Wedemeier.
• Wehn.
• ‡Weidler, 66.
• Weigel.
• Weinhold.
• Weise.
• Wells, 55, 64.
• Welpius.
• Welsh, 73.
• Wencelaus.
• Wendler.
• Wentz.
• Werz.
• Wesellow, 36.
• Westerkamp.
• Weston, 68, 73.
• Wetterhold.
• Whiston, 73, 74.
• White, 93, 97.
• Whiting.
• Wicks, 97.
• Widebergius.
• Widman, 101.
• Widmann.
• Wiedeberg.
• Wiedemann.
• Wiesiger.
• Wigan.
• Wilborn.
• Wilcke.
• Wilder, 74.
• Wildvogel.
• Wilhelmus.
• Willemson.
• Willesford.
• Williams.
• Williamson, 77.
• Willich.
• Willichius.
• Willsford. [Pg 124]
• Wilson, 69.
• Windssheim.
• Wing, 53.
• ‡Wingate, 42, 44, 48, 59, 73.
• Winterfeld.
• Wirth.
• Witekind.
• Witt, xxiv. 33.
• Wittmer.
• Wochenblatt.
• Wohlgemuth.
• Woit.
• Woldenberg.
• Wolff, 68, 70.
• Wolphius.
• Woltmann.
• Woodward, 97.
• Wördenmann.
• Worley.
• Worsop.
• ‡Wright, 69, 97.
• Wucherer.
• Wurffbain.
• Wurster.
• Wurstisius.
• Wybard, 43.
• Xenocrates.
• Xylander, 21.
• Youle, 86.
• ‡Young, 89, 90, 93.
• Zaake.
• Zabern.
• Zacharias.
• Zahn.
• Zaragosa.
• Zell.
• Zeller.
• Zeplichal.
• Zesen, Van.
• Zinge.
• Zoëga.
• Zon.
• Zonsen.
• Zubrod.
• Zucchetta.
• Zurner.

THE END.

LONDON:
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