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ARITHMETICAL BOOKS. 
 
LONDON: 
 PRINTED BY ROBSON, LEVEY, AND FRANKLYN, 
 Great New Street, Fetter Lane. 
 
ARITHMETICAL BOOKS 
 
 FROM 
 
 THE INVENTION OF PRINTING TO THE 
 PRESENT TIME 
 
 BEING 
 
 BRIEF NOTICES OF A LARGE NUMBER OF WORKS 
 
 DRAWN UP FROM ACTUAL INSPECTION 
 
 BY 
 
 AUGUSTUS DE MORGAN 
 
 OF TRINITY COLLEGE, CAMBRIDGE 
 SECRETARY OF THE ROYAL ASTRONOMICAL SOCIETY: FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY 
 AND PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON. 
 
 —— 
 
 ‘* Much surprised, no doubt, would the worthy man have been, had any one told him that two 
 hundred years after his death, when no man alive would think his ideas on the nature of mathe- 
 matics worth a look, the absence of better materials would make his list of” arithmeticians ‘ not 
 only valuable, but absolutely the only authority on several points.”"—DuBLIN REvIEW, No. XLI. 
 
 LONDON 
 TAYLOR AND WALTON 
 
 BOOKSELLERS AND PUBLISHERS TO UNIVERSITY COLLEGE 
 
 28 UPPER GOWER STREET 
 
 1847 
 
Digitized by the Internet Archive 
 in 2022 with funding from 
 University of Illinois Urbana-Champaign 
 
 https://archive.org/details/arithmeticaloookKOOdemo 
 
Cop 2. MATHEMATICS 
 LIBRARY, 
 
 TO THE 
 
 VERY REVEREND GEORGE PEACOCK, D.D. 
 
 DEAN OF ELY, LOWNDEAN PROFESSOR, 
 &c. &c. &c. 
 
 MY DEAR SIR, 
 
 Ir 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* upon any subject with which the name of one indi- 
 vidual 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 friend- 
 ship on your part. 
 
 I have rather grown than made this catalogue. It never oc- 
 curred 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 de- 
 scribed from his own inspection. Knowing, from sufficient experi- 
 ence, the general inaccuracy and incompleteness of scientific lists, I 
 therefore determined to do what I could towards the correction of 
 both, by describing as many works as I could manage to see. From 
 
 * Had the regulations of the work in which it appeared permitted, it would have 
 been most peculiarly appropriate to have inscribed my treatise on the Calculus of 
 Functions to my friend Mr. Babbage. 
 
ll PREFATORY LETTER. 
 
 the Royal Society’s library, the stock of Mr. Maynard the mathe- 
 matical 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, Encyclopzedia 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 pre- 
 sented with that derived from other sources. That plan has been, 
 to attempt some rectification of the numerous inaccuracies of ex- 
 isting 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. 
 
‘ PREFATORY LETTER. lll 
 
 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 primd 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,gand 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 CoLLece, Lonpon, 
 April 29, 1847. 
 
*,* 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 Zntroduction, where he is told to con- 
 sult the Index; and the Additions, where he is told to consult the 
 Introduction. 
 
INTRODUCTION. 
 
 on Oe 
 
 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 in- 
 quiry 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 pre- 
 ponderance of German names arises partly from the number of 
 works written on the subject in Germany being very consider- 
 able, partly from the bibliographical catalogues of that coun- 
 try 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 Rogeg’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 arith- 
 metic published in Latin, French, German, Dutch, Italian, 
 
vi INTRODUCTION. 
 
 Spanish, and English, up to the middle of last century, can- 
 not 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 pre- 
 servation 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 New- 
 ton 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 excep- 
 tions 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 intel- 
 ligence of the sixteenth century had been sufticient to verify 
 and receive the opinion of Copernicus at once, some prede- 
 cessor 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 se- 
 condary 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 occur- 
 rences. 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 ac- 
 count 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 historian has at his command a readier access to 
 
INTRODUCTION. vil 
 
 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.* 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, &ec., and several Taylors, 
 Butlers, &e., 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, Dohren and Diithren. 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 read- 
 ing 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. 
 soon. These obvious mistakes of course have not been ad- 
 mitted. 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 
 Budeeus. Many foreign writers, Heilbronner among the rest, 
 have turned Bucleus into Budeus, so that in all probability the 
 second of these names is a mistake for the first. And yet 
 there may be also some Budeeus who has written on arith- 
 metic; though, having excluded writers on weights and mea- 
 sures 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. 
 
 I have not attempted to translate the names of those who 
 
 * It is necessary, for instance, to keep close watch upon a writer who 
 introduces among his English authors Gul. de Cavendy dux de Xeucathle. 
 
Vill INTRODUCTION. 
 
 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 pos- 
 terity; 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 suc- 
 cessors cite him. It is well to know that Copernicus, Dasy- 
 podius, Xylander, Regiomontanus, and Clavius were Zepernik, 
 Rauchfuss, Holtzmann, Muller, and Schliissel. 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 fini- 
 cally correct than Horonce Phine, which has great antiquity 
 in its favour? In the case of Vieta, Viéte is certainly wrong: 
 he was called by his contemporaries Viet (which I suspect to 
 be derived from the Latin form) and de Viette, never Viéte. 
 
 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 feel- 
 ing 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. 
 
INTRODUCTION. 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 mate- 
 rials 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 arith- 
 metical 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 repre- 
 sentations of informal titles, which, though not originally in- 
 tended for more than sufficient indication of the book men- 
 tioned, 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 as- 
 suming their place in the annals of the knowledge to the 
 progress of which they have contributed. Take the library 
 of the British Museum, for imstance, 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. No- 
 body 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 under- 
 taking. And it would perhaps be difficult to make any regu- 
 lation, under which persons not belonging to the institution 
 might have access to see what is there. Nor will the pub- 
 lication of the catalogue do much towards supplying the de- 
 
 b2 
 
x INTRODUCTION. 
 
 fect. Titles of books are but vague indicators of their con- 
 tents; and a catalogue* 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 ex- 
 amine 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 arith- 
 metic, 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 autho- 
 rity, 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 
 ave 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 rea- 
 sons. 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 is, on the whole, and 
 
 * When the great catalogue of the Museum is published, those who can 
 give house-room to forty or fifty volumes, and time enough to their exami- 
 nation, may have some of the advantage which they would derive from 
 actual access to the books themselves. And those known to be engaged in 
 research will derive a still larger portion of the same advantage from the 
 readiness with which the officers of the Museum will go out of the usual 
 routine of duty to help them. 
 
INTRODUCTION. , xi 
 
 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 unmistakeably 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 oppor- 
 tunity of re-examining while the sheets were in proof. Imi- 
 tation of type [ have not attempted. ‘To have given the full 
 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 soon. 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 
 siz 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 
 
xi INTRODUCTION. 
 
 lies between two leaves of another. But in the sixteenth cen- 
 tury, and even later in Italy, it was common enough to print in 
 guire-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, 123 °7, 8, 
 9, 10. Ifa book were printed in this way, it would certainly 
 be a folio, if the four double leaves of any one quire or gather- 
 ing 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 A, on page 1, A, on page 3, A, on page 
 5, and A, 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: 
 
 (a) The words folio, quarto, octavo, duodecimo, decimo-octavo, 
 refer entirely to size, as completely as in a modern sale-cata- 
 logue, the maker of which never looks at the inside of a book 
 to tell its form. All the very modern distinctions of imperial, 
 royal, crown, atlas, demy, &c. &c. &c. I have relinquished to 
 paper-makers and publishers, who alone are able to understand 
 them. But in old books, the reader must expect. to see the 
 several sizes, each one of them, smaller than in the modern 
 books. When the work is decidedly small of its name, I have 
 noted it by the word ‘small.’ 
 
 (b) When the single word occurs, without any thing more, 
 the signatures are as in the genuine meaning of the word. 
 Thus, as to signatures, folio has two leaves to one letter of 
 signature, guarto four leaves, octavo eight, duodecimo twelve ; 
 and of course double the number of pages. 
 
 (c) When the modern word occurs with the addition of in 
 twos, or in threes, &c. the addition expresses the number of 
 double leaves which belong to one letter of signature: and 
 which I believe would be found, if the books were taken to 
 pieces, to be im each quire or gathering. Thus, folio in ones, 
 or quarto in twos, or octavo in fours, or duodecimo in sizes, 
 would in each case be unnecessary repetition; for the first 
 word, when alone, is intended to express the gathering in the 
 third. But folio in twos would mean the folio size with two 
 double leaves in one quire, folio in fours with four double 
 leaves. Thus, a book of the octavo size, with the quarto sig- 
 
INTRODUCTION. xii 
 
 natures, is octavo in twos: had it been larger, I should have 
 called it guarto. 
 
 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 stance, 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 de- 
 termining whether the book he has in his hand be the one I 
 describe or not. 
 
 This is no unnecessary excess of description. For so fre- 
 quently, 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 de- 
 scribed. Were I to begin this work again, I would in every 
 instance make a reference to some battered letter, or defect of 
 hneation, or something which would be pretty certain not to 
 recur in any real reprint. Ordinary errata would not be con- 
 clusive: 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 duo- 
 decimo. 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 Ste- 
 vinus, an undoubted folio, has thick waterlines both ways. 
 
 6. In the smaller type are entered such remarks as sug- 
 gested themselves on the manner or matter of the work, or on 
 any point arising out of it. In some cases these have ex- 
 tended themselves to short dissertations, such as, on the geo- 
 metrical 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 genu- 
 ineness 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. 
 
 The principal point on which I have distinguished one work 
 
X1V INTRODUCTION. 
 
 from another, is as to the use of the old or new method of per- 
 forming division, which, more than any other single poimt, 
 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, Pea- 
 cock, p. 433. 
 
 With regard to the works themselves, I have made no se- 
 lection, 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 computa- 
 tion, 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, Dechdles, Heilbronner, Scheibel, Kastner, 
 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 in- 
 dexed 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, how- 
 ever, 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 stat- 
 ing 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 Re- 
 giomontanus. And again, we find him, though the editor of 
 an edition of Oughtred, balancing as to which was written first, 
 
INTRODUCTION. XV 
 
 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 sub- 
 ject. 
 Claude Francis Milliet Dechdles was a Jesuit, who pub- 
 lished, in four very large folios, a complete course of mathe- 
 matics, 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 Dechales 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 T'onstall ; 
 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 cen- 
 tury, 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 dis- 
 sertation 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 tor further research. 
 
 Scheibel (Additions) may be considered as partly repetition, 
 partly extension, of Heilbronner, He is one of those biblio- 
 graphers who collect from various sources the names and dates 
 of more editions than those who know catalogues will readily 
 believe in. 
 
 Kistner (Additions) falls under the followmg censure from 
 Dr. Peacock: ‘“‘The meagre sketch which Kastner has given 
 of some insulated works on the subject, generally contrives to 
 omit almost every particular which is essentially connected 
 
Xvi INTRODUCTION. 
 
 with the history of the progress of the science.” This is well 
 merited, inasmuch as the author chose to call his work a his- 
 tory, 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 Kistner his. But, as a bibliography, 
 it may be urged in defence, that he gave fuller descriptions of 
 books than his predecessors. Scheibel, Heilbronner, and De- 
 chales sink into title-writers (and bad ones) before Kastner. 
 
 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 Philo- 
 sophy 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 know- 
 ledge 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 m 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 inves- 
 tigator. 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, contain- 
 ing merely a few scanty precepts,’ with the summary of the 
 contents of that work in the Penny Cyclopedia, article Viga 
 Ganita. Leslie also generalises most fearfully every now and 
 then. He informs us that it was the practice throughout Hurope 
 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 ex- 
 presses the truth in saying that, before the invention of printing, 
 the practice of writing memorial verses was common, as ap- 
 pears by manuscript libraries. It is needless to say that, had 
 the practice of using them been common, the presses of the fif- 
 teenth 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. 
 
INTRODUCTION. Xvi 
 
 Delambre (page 84) wrote on the history of Greek Arith- 
 metic 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 | 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 :* 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 caybe 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 the tunnel. Of arithmetic in parti- 
 
 * It was once intended to publish these treatises separately. Nine years 
 ago, the proprietors of the Encyclopedia Metropolitana so fully intended 
 to publish separately, that they considered themselves aggrieved because I, 
 who had written the mathematical article on, probabilities, wrote a popular 
 work with that word in the title-page, which they alleged, through their 
 agent, was in effect a republication of the former work, &c. Not being 
 able to get them either to litigation or arbitration, I was obliged to write 
 a pamphlet to prove that the charge was frivolous. The pamphlet is un- 
 answered, and all the treatises unpublished (separately) to this day. The 
 latter L regret, for the sake of science. It is a great pity that. Sir John 
 Herschel’s treatises on light and sound, Dr. Peacock’s arithmetic, Mr. 
 Airy’s tides, &c. are thus locked up. 
 
 Cc 
 
XViil INTRODUCTION. 
 
 cular, 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 Hastern 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-traimed palzographist. 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 know- 
 ledge 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 imstance, just what it 
 might have been if the authors had been Persians. or Saracens. 
 How the communication took pce 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 author- 
 ship 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 for- 
 -mer wrote themselves: but Iam 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 ele- 
 mentary 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 called, was the first who wrote on or in the new system, 
 
INTRODUCTION. XIX 
 
 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 descrip- 
 tion. Those who have searched into the matter, merely with 
 a view to Arabic zumerals, 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 arith- 
 metic. 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, 
 hhas received notice and discussion from Mr. Hallam, from a 
 learned writer in the Archzeologia, 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 ques- 
 tion 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 Ita- 
 lian 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 pre- 
 sent themselves. Their only common point is the use of the 
 Arabic or Indian numerals and method of notation. 
 
XX INTRODUCTION. 
 
 The first, or algoristic system, as we may call it, proceeded 
 by systematic rules to the performance of questions of com- 
 putation. 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 arith- 
 metical, 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: buc there is 
 an obvious leaning to the idea that they ought to be so. Geo- 
 metry 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 expres- 
 sion of numerical ratios, I may refer the reader to the article 
 “‘Numbers, old appellations of,” in the Supplement to the 
 Penny Cyclopedia. 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 mentioned by Dr. Peacock, who could not get further 
 
INTRODUCTION. XXi 
 
 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 atten- 
 tion 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 six- 
 teenth 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 commer- 
 cial 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 excep- 
 tions 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 seven- 
 teenth century, and the instantaneous appreciation of the value 
 of the discovery of logarithms, were the result of that supe- 
 riority 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 sub- 
 jects 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 arithiffeticians to prove their rules: and the 
 very word proof, in that science, never came to mean more* 
 than a test of the correctness of a particular operation, by re- 
 versing the process, casting out the nines, or the lke. As 
 
 * At first I had written “degenerated into nothing more;” but this is 
 incorrect. The original meaning of the word proof, in our language, is 
 testing by trial. 
 
 c2 
 
XXil INTRODUCTION. 
 
 soon as attention was fairly diverted to arithmetic for commer- 
 cial 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 cen- 
 tury, 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 arith- 
 meticians 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 arithmeticia s 
 and algebraists had another difficulty to contend with: thei 
 fear of their own half-understood conclusions, and the cau- 
 tion 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-+-ace by two mul- 
 tiplications, instead of using a(d+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 multi- 
 pler were 8 or 7. I haveseen this often enough; and things 
 nearly as strange times without number. But what shall we 
 say to the following; a most sufficieat recommendation of the 
 study of old works to the teacher, aSshewing that the difficul- 
 ties which it is now (I speak to the feacher not the rule-driller) 
 his business to make smooth to the youngest learners, are pre- 
 cisely those which formerly stood in the way of the greatest 
 minds, and sometimes effectually stopped their progress. Per- 
 haps no man of his day had so much power over mathematical 
 language as Wallis. But the following extract ofa letter from 
 
INTRODUCTION, Xxili 
 
 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 24/3, instead of 
 12, not having then met with any example of a number so 
 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 com- 
 mensurable surds, that I resolved to use it for my own occa- 
 sions, 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 per- 
 suade them that the difficulties of progress are of the same 
 character in all ages, will find in old works plentiful and use- 
 ful 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 de- 
 noting the meaning of each decimal place. Dr. Peacock men- 
 tions 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 aprocess, to be useful in pointing out after- 
 wards how another process is to come on, or language is to 
 
XXIV INTRODUCTION. 
 
 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 deci- 
 mal 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 pur- 
 pose: 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.”? Unfortu- 
 nately Napier has not given us instances enough to tell us whe- 
 ther 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 per- 
 manent 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 multiply- 
 ing 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: It gives 123436,, 
 
INTRODUCTION. XXV 
 
 not it is 123,45. This isa 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 thin’ 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 diametri- 
 cally 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 pos- 
 sible claimants. And first, as to Briggs. He does not indeed 
 arrive at the simple decimal point (which is a strong presump- 
 tion 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 abbrevia- 
 tion being common, that Briggs explains his notation as quite 
 peculiar to himself. In the preface, or dd 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 hec humanissime Lector, te admonendum 
 censuil. 1. Si numerus occurrat cui linea subscribitur, notze 
 illee quee supra lineam sunt descriptee, Numeratorem partium 
 constituunt, quarum Denominator semper intelligitur unitas, 
 cum tot Shige quot sunt notee superius posite. ut 75 de- 
 
 signant 57%, vel 3. sic 5 9321 designant 554%)5.” 
 
 Oughtred Lah both the vertical and sub-horizontal 
 separators, thus shutting up the numerator in a semi-rectan- 
 gular 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 sepa- 
 rately, yet it appears in the collection of his works. I have 
 what is called the second edition of this collection, London, 
 
XXVI INTRODUCTION. 
 
 sixteen-thirty-six, guarto, which, ‘for various reasons, I suspect 
 to be the first, thinking that the announcement of second edi- 
 tion merely refers to the fact of its having been previously 
 published during -Gunter’s life. After the jfinis, 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* came 
 out in sixteen-twenty-three; and this, though it makes refer- 
 ence to that work, or at least to its matter. At the beginning 
 of this ‘Generall use, &c.’? common fractions are tsed, even 
 when ‘the denominators are decimal. -At page 13, Briggs’s no- 
 ‘tation appears, without explanation: and 116 04 is the third 
 proportional to 100 and 108; ‘this continues through page 14. 
 In page 15, a dotis added to Briggs’s notation in one instance : 
 100/. in 20 years at 8 per cent becomes 466.0951. At the 
 bottom of this page Briggs’s notation disappears thus, “It ap- 
 peareth before, that 1007. 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 85/. 733: then adding these two together, 
 
 wee have 178/. 326 for the present worth of 100. pound an- 
 nuity for 2 yeeres and so forward.’ After this change, thus 
 made without warning+ in the middle ofa 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 loga- 
 rithms 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 explana- 
 tion the fraction is not thus written, but described as parts, 
 Thus, 32.81 feet in operation is 32 feet 81 parts in the de- 
 seription of thé 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 123/456. To 
 
 * A printed title-page has 1623; ‘an additional engraved and orna~ 
 mented title-page has 1624, 
 
 + As far as I can find, the words cosine and cotangent (which Gunter 
 introduced) appear in the same manner, without warning or explanation. 
 
INTRODUCTION. XXVIII 
 
 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 
 whieh 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. 1 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 funda- 
 mental 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 ex- 
 pression. Nothing is more clear than that the Greek geome- 
 ters, 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 reflect- 
 ing mind can have. 
 
 In looking over the list of books, it will be observed, that 
 during the eighteenth and nineteenth centuries there isa great 
 preponderance of English writers. The law I imposed on 
 myself, of entermg no books which I could not see, neces- 
 sarily 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 able, to 
 
XXVill INTRODUCTION. 
 
 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 1s 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 ap- 
 proximation 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. 
 
 On looking at the fourth edition of Gunter’s work (sixteen- 
 sixty-two), I find that some liberties have been taken with the ori- 
 ginal text: among others, Briggs’s notation is restored in several 
 places, though not entirely. It should also be noticed that the 
 point which generally occurs after the first figure of a logarithm 
 is not a fractional separator, but only divides one integer from the 
 rest. 
 
A LIST 
 
 OF 
 
 WORKS ON ARITHMETIC, 
 
 THEORETICAL AND COMMERCIAL, IN CHRONOLOGICAL ORDER. 
 
 *,* The letter P. followed by a number, refers to the page of Dr. Peacock’s treatise 
 in which the work or its author is mentioned. The brackets [ ] enclose state- 
 ments which I have taken from others, and cannot therefore affirm from per- 
 sonal inspection. For O and N see those letters in the Index. 
 
 Florence, fourteen-ninety-one. Philip Calandri. < Phi- 
 lippi Calandri ad nobilem et studiosum Julianum Laurentii 
 Medicem de Arimethrica opusculum.’ Octavo (small). 
 
 This book, which very few have mentioned at all, and fewer still 
 from inspection, is a part of the rich bequest of the late Mr. Gren- 
 ville to the nation. It begins with a picture of Pythagoras teaching, 
 headed ‘ Pictagoras Arithmetrice introductor.’ In the preface is 
 the following on Leonard of Pisa: ‘ Vero e che il modo del notare e 
 numeri con decte figure dice Lionardo pisano haver nel Mcc. incirca 
 rechato dindia in Italia: et decti carateri: o vero figure essere 
 indiane : et appresso deglindi havere imparato la copulatione desse.’ 
 He explains the leading rules, except division, for integers and for 
 lire, soldi, and denari. Division he puts down in examples, and 
 appears to have mistaken the mode of working, or to have had an 
 incorrect printer. His notion of a divisor is curious. When he 
 divides by 8, he calls his divisor 7; demanding, as it were, that 
 quotient which, with seven more like itself, will make the dividend. 
 He also describes the rules for fractions, and gives some geometrical 
 and other applications. The book is in black letter, but the nume- 
 rals are in a thin and small-bodied type. At the end of the book is, 
 ‘Impresso nella excelsa cipta de Firenze per Lorenzo de Morgiani 
 et Giovanni Thedesco da Maganza finito a di primo di Bennaio 
 1491.’ 
 
 B 
 
A CHRONOLOGICAL LIST OF 
 
 Venice, M.cccc.|xliiii., fourteen-ninety-four. Lucas Pa~ 
 cioli, de Burgo Sancti Sepulcri. ‘Summa de Arithmetica 
 _ Geometria Proportioni et Proportionalita.’ Folio in fours. 
 
 Tusculano, fifteen-twenty-three. Lucas Pacioli. ‘Sum- 
 ma 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* finissimi et mirabil colone 
 marmorei: inumeri fragmenti di alabastro porphidi et ser- 
 pentini. Cose certo lettor mio diletto oculata fide miratu digne 
 sotterra se ritrovano.’ Folio in fours (ON). 
 
 These are the full titles of the two editions of this celebrated 
 work. Both editions are by one printer, Paganino of Brescia, and 
 are beautifully printed : the type of the second being a good instance 
 of the black letter in its state of approach to what is now called Ro- 
 man letter. The work itself has been described by Hutton, Mon- 
 tucla, Peacock, Libri, &c.; but it would yet require a volume of 
 description to do it justice. It is sometimes called the first work on 
 arithmetic printed ; but Calandri, Peter Borgo, three already men- 
 tioned in the Introduction, and perhaps more, take precedence. But 
 it is certainly the first printed on algebra, and probably the first on 
 book-keeping. P. 414, 424, 429 &c., 432 &c., 451, 460, 462; Hut- 
 ton, 7racts, vol. ii. p. 201. 
 
 On comparing my copy of the first edition with that in the British 
 Museum, I found one of those phenomena which so frequently 
 occur in very old printed books. The first leaves of the two copies, 
 to the number of about thirty, and the first leaf of the geometry, 
 are not from the same setting up. The endings of the pages, and 
 the ornaments of the capital letters, are different in the two. Nor 
 does either of them agree with the second edition. A part of the 
 first impression may have been lost, so that a second setting of 
 types was required to replace that part. Several mathematical biblio- 
 graphers of note enter Pacioli as Lucas di Borgo, and even as Bor- 
 godi, Lucas, and some do not mention the first edition. 
 
 The Latin Pactolus is usually spelt in Italian Paccioli, but Libri 
 spells it Pacioli. I believe that the various assertions that Pacioli 
 wrote Euclid in Jéalian, arise from the geometry of the work above 
 described being Italian: but it is not Euclid, though of course 
 founded on him. 
 
 There is another old book of which I have reserved the men- 
 tion till now, because its author has been confounded with Pacioli. 
 Brunet gives it as ‘La Nobel opera di Arithmetica . . . compilata 
 per B. [should be P.] Borgi,’ Venice, fourteen-eighty-four, quarto : 
 and Hain has ‘ Borgo (Pietro) Venet. Aritmetica,’ Venice, fourteen- 
 
 * Cns in the title. 
 
WORKS ON ARITHMETIC. 3 
 
 eighty-two, quarto. This edition Hain had not seen: but he had 
 seen the next (the one mentioned by Brunet), which begins ‘ Qui 
 comenza la nobel opera di arithmetica . . . per Piero Borgi,’ Venice, 
 fourteen-eighty-four, quarto. And he gives two other editions, 
 Venice, fourteen-eighty-eight and ninety-one, quarto. Accord- 
 ingly, Mattaire sets down the first edition of Lucas di Borgo as of 
 1484, and Dr. Peacock adopts this date. It is quite certain that 
 Dr. Peacock must have had one of the editions of Pacioli before him 
 when he wrote; and we are therefore to suppose that, in stating the 
 date of the first edition, he followed Mattaire, and presumed an edi- 
 tion earlier than any he had seen. Tartaglia refers to Peter Borgo. 
 P. 458. 
 
 No piace, no date. John Muris. ‘ Arithmetices Com- 
 pendium ex Boetii Libris per Johannem Muris excellentis in- 
 genii virum Accurate congestum.’ Quarto. 
 
 This is a different edition from the one presently mentioned. 
 John Muris seems to have escaped the notice of bibliographers as an 
 arithmetician. I think I remember that there is in Hawkins’s His- 
 tory of Music some discussion of his ideas on the musical scale, {on 
 which was published his work, Leipsic, fourteen-ninety-six, folio]. 
 Muris lived before the invention of printing, but when I cannot as- 
 certain. [His astronomical tables are preserved in manuscript. ] The 
 present tract has twelve leaves unpaged, and is certainly of the very 
 earliest part of the sixteenth century, if not of the fifteenth. 
 
 Venice, fifteen-one. Geo. Walla. ‘De expetendis et 
 fugiendis rebus opus.’ Folio in fours. 
 
 The first part of this is ‘De Arithmetica libri iii, ubi quedam a 
 Boetio pretermissa tractantur’ (O). Certainly he supplies some of 
 the omissions of Boethius; the four rules for instance. The work 
 was published by the son, Joh. Pet. Valla. I have seen an earlier 
 edition mentioned, but I cannot find any trustworthy account of 
 such a thing. 
 
 Cologne, fifteen-one. John Huswirt. ‘ Enchiridion No- 
 vus Algorismi summopere visus De integris. Minutiis vul- 
 garibus Projectilibus Et regulis mercatorum sine figurarum 
 (more ytalorum) deletione percommode tractans,’ &c. Quarto, 
 twos and threes (O). 
 
 A short treatise, apparently one of the earliest printed in Ger- 
 many on the Arabic system, The projectilia are counters. The 
 rules are verified by casting out the nines. The words ‘sine figu- 
 rarum deletione’ are not made good in the rule of division. On this 
 work see the Companion to the Almanac for 1844, p. 3. 
 
 Paris, fifteen-three. Joh. Faber Stapulensis, Jodo- 
 cus Chlichtoveus, and Carolus Bovillus. ‘In hoc li- 
 
4 A CHRONOLOGICAL LIST OF 
 
 bro contenta Epitome compendiosaque introductio in libros 
 Arithmeticos divi Severini Boethii,’ &c. &e. Folio in fours. 
 
 This book contains an epitome of Boethius by Faber of Etaples ; 
 with a commentary and an arithmetical collection of rules by Chlich- 
 toveus, his pupil; a compendium of geometry, a book on the quadra- 
 ture of the circle, and cubication of the sphere, and a book on perspec- 
 tive, by Charles Bovillus; with an astronomical compendium by Faber 
 already mentioned. This book is one of the earliest printed by Henry 
 Stephens and his partner Wolfgang Hopilius. It is the first edition, 
 Thave no doubt, of Faber’s epitome of Boethius, though Heilbronner 
 and Murhard assert the contrary, perhaps misled by Faber’s edition 
 of Jordanus. As to the contents, the Arithmetic of Boethius was 
 the classical work of the middle ages. It consists of statements of 
 the commonest properties of numbers, under a great many classifica- 
 tions, to each of which a name is given. ‘The second work, of arith- 
 metical rules, shews the very low state of the art. It takes pages 
 upon pages to explain the simple rules, though no examples are 
 ventured on which have more than three figures. Montucla says 
 that the quadrature of Bovillus was only saved from the laughter of 
 geometers by its obscurity. Buta work printed by Henry Stephens, 
 and containing Boetius and Faber, must have been very far from ob- 
 scure in itsday. The historians of mathematics confine themselves 
 to works the reputation of which has lasted: but they ought not to 
 make the state of their own minds, with respect to the rest, a cri- 
 terion of that of the contemporary readers. 
 
 Basle, fifteen-eight. Gregorius Reisch. ‘Margarita 
 Philosophica, cum additionibus novis: ab auctore suo studio- 
 sissima revisione tertio superadditis.’ Quarto in fours. 
 
 According to Kloss (from his own copies) the first edition of this 
 curious book is Friberg fifteen-three, the second and third both Stras- 
 burg fifteen-four (one printed by Gruninger, which I have seen, and 
 one by Schott), and he calls the one before me the fourth. But 
 Hain marks as the first edition one which appears to be printed at 
 Hleidelberg, fourteen-ninety-six, quarto. The one before me is not 
 the third; it is evident that a former title-page has been reprinted : 
 for though this edition is printed (as appears from the colophon) by 
 Furter and Scotus of Basle, the title-page bears ‘ Jo. Schottus Argent. 
 lectori S.’. Which agrees with Kloss’s statement that the previous 
 edition to this one was printed by Schott of Strasburg. The Arith- 
 metic (O), which is a part of the system of philosophy here laid down, 
 has a frontispiece representing Boethius at one table with Arabic nu- 
 merals before him ; and Pythagoras at another with counters. Pytha- 
 goras among the Greeks, Apuleius and Boethius among the Romans, 
 were often made the inventors of arithmetic. The arithmetic is di- 
 vided into speculative and practical. The former is a summary of 
 Boethius, often in the words of John de Muris. The latter is a short 
 treatise on Algorithm, as it was called, or the rules of computation 
 by the Arabic numerals.. There is also computation by counters, 
 
WORKS ON ARITHMETIC. 5 
 
 fractions common and sexagesimal, and the rule of three. Many 
 works of fifty years later do no more difficult questions. That Boe- 
 thius was the author of the Arabic numerals was a common notion 
 at the time, revived in our own day. I have seen another edition 
 of Strasburg, fifteen-twelve; and there is said to be another edition 
 by Orontius Fineus, Paris (?) fifteen-twenty. There is also an 
 Italian translation, Venice, fifteen-ninety-nine, quarto, with the addi- 
 tions of Orontius, translated by Giovan Paolo Galucci. 
 
 If the number were sufficient of those who wish to take their 
 notions of liberal education in Europe at the time immediately pre- 
 ceding the Reformation from original sources, and not from the re- 
 ports of others, a reprint of the Margarita Philosophica would be 
 made. ‘The diversity of the matters which it treats, and the large- 
 ness of its circulation, stamp it as the best book for such a purpose. 
 
 The Margarita Philosophica is the earliest work I have found 
 in which mention is made of that peculiar system of measures 
 which was current among the mathematicians of the sixteenth cen- 
 tury, and which has caused no little confusion among writers on 
 metrology. I have already given some account of this ill-understood 
 system, and shall here endeavour to present the whole case with some 
 additional evidence. 
 
 The Roman foot (of 11°62 inches English) was of course esta- 
 blished throughout the empire: and with it the Roman pace* of 
 five feet, or 58°1 inches or 4°84 feet English. The natural pace of 
 a man in our day is as nearly as possible five English feet: Pauc- 
 ton’s experiments on the walking paces of individuals gave him 59-7 
 inches English, or 4°98 feet. In the British army the step, both 
 what is called the ordinary step and the quick step, is, by regula- 
 tion, thirty inches: making a pace of five feet. The Roman pace, 
 by which distances were actually measured, was that of a soldier on 
 the march: and, as might be expected, the weight of his arms and 
 other equipage seems to have shortened his pace a little. But 
 so near were these measures, as actually used by the Romans, 
 to the natural ones from which they derived their names, that it 
 was customary, not only to recur to legal standards, but, in the 
 absence of ready access to them, to make use of the natural foot 
 and pace. And we also know from Roman writers, that a some- 
 what fanciful relation, though not very far from the truth, was 
 established between the breadth of the hand across the middle of 
 the fingers, the palm, and the length of the foot. It was taken 
 that four palms made a foot: and a palm was made of course to 
 consist of four (average) finger-breadths or digits. ‘This division 
 into palms and digits was the most recognised division of the 
 Roman foot: that into inches, or wnci@, is well known not to be- 
 long to the foot merely, but to any thing else. Whatever magni- 
 tude was called unity, the wncia was its twelfth part. Accordingly, 
 all the Roman foot-rules which have been found in ruins or excava- 
 
 * The pace is derived from the double step, being the distance from the extremity 
 of the heel at the place from which it is removed in walking to the same at the place 
 in which it is set down again. 
 
 B 2 
 
6 A CHRONOLOGICAL LIST OF 
 
 tions have the digital division, to which some (most, I believe) have 
 the uncial division superadded. All this I take to be too well esta- 
 blished to require the citation of any authorities. 
 
 It is quite out of my power, or, as far as I know, of that of any 
 one else, to trace the gradual alteration of the foot in different coun- 
 tries. It does not appear that any means were taken to institute 
 comparisons of various measures, to be preserved as public records. 
 Such means could have been found: that which was then the com- 
 mon church of Christendom might have easily regulated the weights 
 and measures of Europe, even without appearing to do so. But in 
 all probability, the extent of the variations was not well known until 
 it was too late. We know, however, as a fact, that the geometers were 
 successful in establishing a measure among themselves, and com- 
 municating it through Europe on paper. ‘This measure, I have no 
 doubt, they believed to be the true Roman foot: for they divide it 
 in Roman denominations, make use of it in their quotations from 
 Roman authors, and never hint at their having any other notion of 
 a Roman foot. And moreover, the writers who, in the sixteenth 
 century, recovered the true Roman foot, never mention any peculiar 
 geometrical foot in use among mathematicians, or in any way dis- 
 tinguish the latter from the wrong Roman foot which they were 
 correcting. That the geometers believed the Roman foot, that is, 
 their own foot, to be the human foot, might be easily proved. And, 
 with such belief, they would make their so-called Roman foot too 
 short. From a hundred measures of the feet of adult men, furnished 
 to me by a boot-maker, and taken as they came in his books, I find 
 the average length of the Englishman’s foot to be 10°26 inches, in our 
 day: or an inch anda third shorter than the Roman lineal measure. 
 
 This geometrical foot of the mathematicians is, I make no doubt, 
 the geometrical foot to which writers of the seventeenth century 
 refer, or mean to refer. But, not long after the true restoration of 
 the ancient measures, there arose a disposition among those who 
 inquired into the subject to seek a mystical origin of weights and 
 measures, on the supposition of some body of exact science once 
 existing, but now only seen in its vestiges: a disposition which is not 
 yet entirely extinct. Some speculated on the pyramids of Egypt, 
 and tried to establish that the intention of building those great 
 masses was that a record of measures founded on the most exact 
 principles might exist for ever. But more turned their attention to 
 the measurement of the earth, and, by assuming nothing more diffi- 
 cult than that a degree of the meridian a thousand times more ac- 
 curate than that of Eratosthenes was in existence hundreds, if not 
 thousands, of years before him, it was easy enough to make out 
 that the whole system of Greek, Roman, Asiatic, Egyptian, &c. 
 measures was a tradition from, or a corruption of, this venerable 
 piece of lost geodesy. There runs through all these national systems 
 a certain resemblance in the measures of length: if a bundle of fag- 
 gots were made of foot-rules, one from every nation ancient and 
 modern, there would not be any very unreasonable difference in the 
 lengths of the sticks. 
 
WORKS ON ARITHMETIC. 7 
 
 The metrologists who treat this subject handle it according to 
 their several theories. Those who have none in particular either 
 neglect it altogether, or speak of its length as uncertain, or define, 
 with Dr. Bernard, the geometrical pace as being five feet of its 
 own kind, without saying what this kind is. Those who have the 
 notion of the old measure of the meridian accommodate it to their 
 supposed ancient measure; but at the same time, those of most 
 research and note make it less than their Roman foot. Thus 
 Paucton makes it nine-tenths of the Roman foot, which, with 
 his version of that measure, is 10°9 inches English, and with the 
 true one, 10°5 of the same. Similarly, Romé de L’Isle makes it 
 more than half an inch (French) less than his Roman foot. As 
 they do not refer to the geometers of the middle ages, I can- 
 not guess whence they get their notion, otherwise than from their 
 theory. . 
 
 I now proceed to demonstrate the existence of this geometrical 
 foot, which I believe to have been the effort of mathematicians to 
 perpetuate and make common what they took to be the Roman foot, 
 on the supposition that it was nothing but the average length of the 
 human foot. 
 
 Passing over the general expressions of writers who refer to the 
 use of the parts of the body in measurement, and who sometimes 
 distinctly state that the determination of the human foot is neces- 
 sarily that of the Roman measure, I take first the statement of Cla- 
 vius, whose term of active life was the latter half of the sixteenth 
 century and who says* very distinctly that the mathematicians, to 
 avoid the diversity of national measures, had laid down a system for 
 themselves. The table of measures which he gives (and dozens of 
 other writers before him) is as follows: 
 
 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 =. 1g = 1 cubit. 
 160= 40=10= 23=12= 1 step (gressus). 
 $20= 80=20=—.5 =3)=2 = ] pace. 
 
 640 =160=40=10 =62 =4=2=1 perch (pertica). 
 125 paces | Italic stadium. 
 8 stadia 1 Italic mile. 
 4 Italic miles a German mile. 
 5 Italic miles a Swiss mile. 
 
 The constant reference to this barleycorn measure (which is 
 seldom, if ever, omitted) induced me to try what it would really 
 make. There is some difference between the breadths of barley- 
 corns. A certain statement of Thevenot (cited in the History of 
 
 * “Enumerande sunt mensure quibus mathematici, maxime geometre, utuntur. 
 Mathematici enim, ne confusio oriretur ob diversitatem mensurarum in variis re- 
 gionibus (quzlibet namque regio proprias habet propemodum mensuras), utiliter ex- 
 cogitarunt quasdam mensuras, que certe ac rate apud omnes nationes haberentur.’ 
 —Comm. in Sacroboscum. 
 
8 A CHRONOLOGICAL LIST OF 
 
 Astronomy, Lib. Usef. Kn.) makes the breadths of 144 grains of orien- 
 tal barley give 14 French feet ; at which rate 64 would only give 8°53 
 inches English. A sample from a London shop gave me (when the 
 largest grains were picked out) 33 to just more than five inches. 
 Some other samples, procured from two different parts of England 
 as the finest which could be got, gave 33 to 5 inches, 33 to 5:1 
 inches, and 33 to 5:1 inches. The average of these is 9°8 inches to 
 64 grain-breadths: a result which coincides more nearly than could 
 have been expected with the following determinations. 
 
 There is a chain of writers who have studied to perpetuate their 
 geometrical foot by causing a line to be laid down on the page, 
 representing the digit, the palm, or the foot. Sometimes the palm 
 or foot is divided into digits: and of course [ rely most on those 
 whose subdivisions are the best. As paper is apt to shrink as it 
 becomes old, the foot deduced from these will be somewhat too small, 
 and it may be afterwards discussed how much it should be length- 
 ened. Leaving this for the present, I give the measurements from 
 different authors. 
 
 Margarita Philosophica, above described. In the Strasburg edition 
 of fifteen-four, the length of the geometrical palm is less than 23 
 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. Stoffler. ‘ Elucidatio Fabrice 
 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. ‘ Monalosphzrium.’ 
 Folio in threes. 
 
 Paris, fifteen-twenty-eight. John Fernel. ‘Cosmotheoria.’ 
 Folio in threes. 
 
 The historical mistake arising out of these works is the most re- 
 markable circumstance attending the lossof the geometrical foot. While 
 Fernel was publishing the first work, he was meditating (or perhaps 
 executing) his famous measurement of a degree of the meridian. In 
 this first workhe lays ‘his geometrical foot down the page, with great 
 care, as he says (omni molimine). Intwo copies of this work which I 
 have examined, the length of the foot is within a sixtieth of an inch 
 of nine inches and two-thirds, giving 9°65 inches. In the second work, 
 in which he announces his measure of the degree, he states that five 
 of his paces and those of men of ordinary stature make six geome- 
 trical paces; which, he adds, is agreeable to the opinion of Cam- 
 panus and others (at least a century and a half before), who made 
 the mile of 1000 common paces to be 1200 geometrical paces. 
 Allowing 60 inches (English) to a common pace, which is rather 
 over than under the truth, this gives a geometrical pace of 50 inches, 
 
WORKS ON ARITHMETIC. 9 
 
 and a foot of ten inches. This is enough to shew that Fernel was, 
 in the second instance, speaking in rough terms of the foot which 
 he printed omni molimine in the first. ‘The geometrical pace being 
 forgotten, and the monalospherium also, the modern historians have 
 assumed that Fernel used the Paris foot: by which he is made to 
 appear to come very near the real degree, whereas he is fifteen miles 
 wrong. 
 
 Paris, fifteen-fifty-two. Jac. Koebelius. ‘ Astrolabii Declara- 
 tio.’ Octavo. 
 
 This worthy astrologer, after referring to the perfect notoriety of 
 the system of measures, gives a digit and a palm. The digit is 
 nine-sixteenths of an inch (English), giving a foot of nine inches. 
 The palm is 24 inches and one-sixteenth, giving 10 inches and a 
 quarter to the foot. The book is small, and the palm incorrectly 
 subdivided. 
 
 Frankfort, sixteen-twenty-one. Peter Ryff. ‘ Questiones Geo- 
 metrice.’? Quarto. 
 
 Four very accordant palms are given, indifferently subdivided 
 into digits. Each palm is 23 inches and three-sixteenths, giving a 
 foot of 9°75 English inches. 
 
 From these different sources, good and bad, we have for the geo- 
 metrical foot 9°8, 9:9, 10°56, 9°75, 9°65, 10, 9, 10°25, 9°75 inches : the 
 mean is9°85. But much the best authorities are Fernel and Stoffler, 
 because they are the greatest names, have given the whole foot, and 
 have taken the greatest pains with the subdivisions. Their results 
 are 9°75 and 9°65, with a mean of 9°7. 
 
 Taking this as the foot on paper, it remains to ask how much it 
 must be lengthened to allow for the shrinking of the paper. At first, 
 relying on the plate in Dr. Bernard’s work on ancient weights and 
 measures, ir which the English foot appears to have shrunk by its 
 42nd part, I was disposed to lengthen the above in the ratio of 41 to 
 42. But observing that an older English foot, figured in the ‘ Path- 
 way to Knowledge,’ 1596, has shrunk only by its sixtieth part, I am 
 rather inclined to consider the shrinking of Bernard’s as an ex- 
 treme case. And moreover, the two copies of the Monalospherium 
 give the same foot within one-hundredth of an inch certainly, and 
 less: and it is very unlikely that if the paper had shrunk much, it 
 should have shrunk so equally in two different copies” But, taking 
 one-fiftieth as the outside, it follows that the geometrical foot is any 
 thing the reader pleases between 9°7 and 9:9 English inches. ‘The 
 result from modern barley* gives 9°8, as above shewn. 
 
 It is remarkable how completely the English writers are in igno- 
 rance of the existence and use of the geometrical foot among their 
 continental neighbours. Blundeville takes Stoffler, in the work above 
 mentioned, to be speaking of a German foot, which, says he, Stoffler 
 makes to be 24 inches less than the English foot. 
 
 * Those who have tried to make the /engths of three barleycorns into an inch will 
 probably think little of this mode of judging. But I observed that in samples of bar- 
 ley of very different apparent fineness, the difference was in the length o/ the corns, 
 the breadths hardly varying at all. 
 
10 A CHRONOLOGICAL LIST OF 
 
 Paris, fifteen-fourteen. Boethius, Jordanus Nemo- 
 rarius, Shirewode (!), J. Faber (Stapulensis). ‘In hoe 
 opere contenta Arithmetica . . . Musica. . . epitome. . 
 Boethii: rithmimachie ludus qui et pugna numerorum appel- 
 latur. Folio in fours. Second edition. 
 
 The first arithmetic is by Jordanus, the second by Boethius, both 
 edited and commented by Faber. The music (a tract on which was 
 in those days little but a tract on fractions under musical names) is 
 also by Faber. The Rithmimachia I suppose to be the work of Shire- 
 wode’s mentioned in the Introduction, but no author isnamed. It is 
 a short triple dialogue on the properties of numbers, with some sort 
 of numerical game. Libri attributes it to Faber, probably from its 
 appearance in this edition without an author’s name, and headed by 
 an address from Faber, the editor. This second edition was printed 
 by H. Stephens [the first, Paris, fourteen-ninety-six, folio, was printed 
 by Hopilius, whose partner H. Stephens afterwards was, under the 
 superintendence of David Lauwxius,* of Edinburgh]. 
 
 Paris, fifteen-fourteen. Wicolas Cusa. ‘ Hee accu- 
 rata recognitio trium voluminum operum Clariss. P. Nicolai 
 Cusee Card. &c.’ In three volumes, folio in fours. 
 
 Cardinal Cusa is put down in several arithmetical lists because 
 one of his opuscula is entitled de Arithmeticis Complementis. But it 
 is not a work on arithmetic, nor does it even proceed by arithmetic. 
 Or it may be that John Cusa, next mentioned, may have been con- 
 founded with Nicolas. For Cusa, see the Companion to the Almanac 
 for 1846, p.14, and Penny Cyclopedia, ‘ Motion of the Earth.’ 
 
 Vienna, fifteen-fourteen. Joh. Cusanus. ~ ‘ Algorith- 
 mus 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). 
 
 Computation by counters and Roman numerals: the Arabic nu- 
 merals are explained, but not used. In the frontispiece is a cut 
 representing the mistress settling accounts with her maid-servant 
 by an abacus with counters. This book is said by Kloss to have 
 been also printed by Kobel himself at Oppenheym in the same year. 
 
 * Capiat qui capere potest is the only general principle on which names like this 
 can be read back into the vernacular. I once found James Hume, well known as a 
 mathematician at Paris in the beginning of the seventeenth century, described as 
 Scotus Theagrius. What part of the land of cakes this might be, would probably have 
 eluded all the books in existence: but a Scottish friend, to whom I mentioned my 
 difficulty, solved it at once, by telling me that he must have been one of the Humes 
 of Godscroft, a place in which it seems certain Humes are; or were, lords of the soil. 
 
WORKS ON ARITHMETIC. hi 
 
 Oppenheym, fifteen-fifteen. Jacob Kobel. ‘Eyn New 
 geordnet Vysirbiich.’ 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, Propor- 
 tiones breves, de latitudinibus formarum, Algorithmus D. 
 Georgii Peurbachii in integris, Algorithmus Magistri Joh. de 
 Gmunden de minuciis physicis.’ Quarto. 
 
 The name of George Tannstetter as editor of this collection is a 
 fair guarantee for the genuineness of the several works. 
 
 The first is an abridgment of Boethius, by John de Muris, of 
 whom elsewhere. 
 
 The second is on proportion by Thomas Bradwardin (of the time of 
 Edward II1.), who is Bradowardinus, Bragvardinus (as in the work 
 before me), Bragadinus (a confusion perhaps between him and Braga- 
 dini), &c. according to the fancy of the speller : Tanner says the book 
 was printed at Paris in 1495. Bradwardine is better known among 
 theologians for his book against Pelagius, than among arithmeti- 
 cians. He begins thus: ‘Proportion is either that which is com- 
 monly so called, or properly so called. Proportion commonly so 
 called is the mutual habitude of two things. Proportion properly 
 so called, is the mutual habitude of two things of the same kind.’ 
 Perhaps the authority of an Archbishop of Canterbury (for he was 
 or was to be nothing less) may induce those who teach the ‘rule of 
 three to remodel their plan according to the archiepiscopal dictum, 
 which is also that of common sense. [Other mathematical works of 
 Bradwardine were printed. | 
 
 The third book is on the areas of figures (though nothing to 
 any purpose is done), with the leading idea that the difficulty arises 
 from variation of breadth. Peurbach’s work is a summary of ope- 
 rations like that of Sacrobosco; but though written to explain the 
 Arabic’ notation, it obviously takes for granted that that notation 
 is generally known. John de Gmunden’s work is on sexagesimal 
 fractions (minuci@ physic@). 
 
 Paris, fifteen-fifteen. Gaspar Lax. ‘ Arithmetica spe- 
 culativa magistri Gasparis Lax Aragonensis de sarinyena duo- 
 decim libris demonstrata.’ Also, same date and place, ‘ Pro- 
 portiones magistri Gasparis,’ &c. Folio in threes. 
 
 This is a very diffuse and extensive work, in small black letter. 
 It treats only of the simple properties of numbers ; though it is evi- 
 dent that to a knowledge of Boethius, Lax added that of the arith- 
 metical books of Euclid. The thing which appears most surprising 
 in this and other works of the same kind, is the apparent difficulty 
 of dealing with numbers. Here are upwards of 250 small black- 
 letter pages in folio, filled with propositions on the simplest properties 
 
12 A CHRONOLOGICAL LIST OF 
 
 of numbers, and the reader looks in vain for any number so high 
 as 100 used by way of exemplification. For any thing that appears. 
 the author could not count as far as 100. Montucla says that Lax 
 was afterwards Pope; but I am assured by a learned Catholic bishop 
 that there never was any Pope of the name. 
 
 Lyons, fifteen-fifteen. John de Lortse (Dominican). 
 “Oeuvre tressubtile et profitable de lart et science de arist- 
 meticque: et geometrie translate nouvellement despaignol en 
 francoys Auquel est demonstre par figure evidemment : tant le 
 nombre entier: nombre rompu : regle de compaignies : soubde 
 fin: que toutes aultres choses qui par geometrie et arist- 
 meticque peuvent estre comprises: comme appert par la table 
 cy apres mise.’ Quarto (O) (166 folios). 
 
 This is perhaps the first book in French on commercial arith- 
 metic. JI have not found mention of it in any catalogue. Itisa 
 good Italian importation, through Spain: and must have made the 
 readers of the Boethian school stare to see what arithmetic really 
 could do. We have thus two Spanish books published at Paris in 
 one year: another (see Siliceus presently mentioned) had been pub- 
 lished there the year before. 
 
 Venice, fifteen-twenty. R.Suiseth. ‘Calculator. Sub- 
 tilissimi Ricardi Suiseth Anglici Calculationes noviter emendate 
 atque revise.’ Folio in threes. 
 
 This man is Richard at the beginning of the book, Raymund at 
 the end, while Tanner [and Gesner] call him Roger. His name 
 seems to have been Swinshead, latinised into Swincetus, Suicetus, 
 Suineshevedus, &c. The title of his book, Calculator, has given 
 rise to the fiction that he was a great arithmetician, and even al- 
 gebraist ; which, as he lived in the fourteenth century, would have 
 made him a very remarkable man. But nothing can be further 
 from the truth: the book is full of philosophy about intension and 
 remission, and siccity, and calidity and frigidity, and other quis-, 
 guilie Suicetice, as some afterwards called them. That number and 
 magnitude are used occasionally, is all that can be said. 
 
 Brucker, according to Enfield, Hallam, &c., says this book is as 
 scarce as a white raven. But probably this refers to one of the pre- 
 vious editions. Hain gives three of the fifteenth century, marked 
 ‘Padue (no date),’ ‘ Papie, 1488,’ and ‘ Papie, 1498.’ All these 
 Suissets are Richards. (See Tanner in verb. ‘ Swincet ;’ Wallis Op. 
 t. iii. pp. 673, 675, 680, 685.) 
 
 Lyons, fifteen-twenty. Stephen de la Roche Ville- 
 franche. ‘ Larismethique nouvellement composee divisee en 
 deux parties dont la premiere tracte des proprietes parfections 
 et regles de la dicte Science,’ &c. Folio in fours (QO). 
 
 A large treatise, very full on commercial arithmetic, and con- 
 
WORKS ON ARITHMETIC. 13 
 
 taining also geometry. I suppose we should find it largely in- 
 debted to the Spanish books above named. 
 
 Paris, fifteen-twenty-one. Boethius. ‘Divi Severini 
 Boetii Arithmetica duobus discreta libris: adjecto commen- 
 tario, mysticam numerorum applicationem perstringente, de- 
 clarata.’ Folio in threes. 
 
 The commentary is by Girardus Ruffus, and for absurdity and 
 dulness, it ought to rank high among the mystic commentaries : 
 even to the justification of Cornelius Agrippa, when he asserts that 
 arithmetic is not less superstitious than vain, adding, like a phi- 
 losopher of his day, that it is only valuable to merchants for the low 
 and mean benefit of keeping their accounts. 
 
 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. 
 
 [There is said to have been an earlier Strasburg edition, but 
 I have not seen it: there are several other editions, but no Eng- 
 lish reprint.| For the life of Tonstall see Anth. Wood, Ath. Oxon. 
 in verb. ‘This book was a farewell to the sciences on the author’s 
 appointment to the see of London (see the preface) : it was published 
 (that is, the colophon is dated) on the 14th of October, and on the 
 19th the consecration took place. This book is decidedly the most 
 classical which ever was written on the subject in Latin, both in 
 purity of style and goodness of matter. The author had read every 
 thing on the subject, in every language which he knew, as he avers 
 in his dedicatory letter to Thomas More, and had spent much time, 
 he says, ad urst exemplum, in licking what he found into shape. 
 The wonder is, that after this book had been reproduced in other 
 countries, and had become generally known throughout Europe, the 
 trifling speculations of the Boethian school should have excited any 
 further attention. For plain common sense, well expressed, and 
 learning most visible in the habits it had formed, Tonstall’s book has 
 been rarely surpassed, and never in the subject of which it treats. 
 It seems to have been very little known to succeeding English 
 writers. P. 419, 426, 427, 439. 
 
 Venice, fifteen-twenty-three. Sacrobosco. ‘Algoris- 
 mus domini Joh. de Sacro Busco noviter impressum.’ Quarto. 
 
 The edition of this work mentioned by Watt, as by Cirvellus, 
 fourteen-ninety-eight, is a mistake: [it was the work on the sphere 
 which Cirvellus edited in that year] (Hain). Dr. Peacock (p. 416) 
 thinks that this work is attributed to Sacrobosco without sufficient 
 
 c 
 
14 A CHRONOLOGICAL, LIST OF 
 
 reason, and mentions it only as a manuscript. Mr. Halliwell. re-. 
 printed it in the Rara Mathematica, evidently under the impression 
 that it had never been printed. If it be the work of Sacrobosco, it 
 establishes beyond doubt that he had the Arabic numerals and the 
 method of local value: being nothing but a summary of rules for 
 that arithmetic. The words noviter impressum are ambiguous: they 
 may either imply a first impression or any succeeding one, though I 
 have commonly found them used in the latter sense. 
 
 I should pause before I rejected as spurious a work which is 
 attributed to Sacrobosco in many manuscript copies extant in dif- 
 ferent countries, which was printed under his name as early as 1523, 
 and is often cited as his. Dr. Peacock lays stress upon there being 
 no mention of the Arabic system in his other works, those on the 
 sphere and on the calendar. But against the presumption drawn 
 from this circumstance, it may be urged that it was not likely 
 he would introduce mention of a new system of arithmetic in 
 works intended for common use, though he might write a separate 
 work to explain that system. And he would have no motive for 
 alluding to a method which his readers were not acquainted with. 
 There is great probability that Sacrobosco was acquainted with the 
 modern system, which all other evidence goes to shew was intro- 
 duced into Italy before his time. 
 
 It is of course very possible, or, looking at the progress of other 
 things, most probable, that isolated individuals had obtained the 
 Arabic notation from the East, or from Italy, practised it, and 
 written in it, long before it obtained the smallest general currency. 
 
 But there is one circumstance which seems to me to lend more 
 than presumption to a still wider supposition, namely, that Arabic 
 notation and rules were known beyond the bounds of Italy, in the 
 time of Sacrobosco, to more than a few isolated individuals. Except 
 by importations from the East, it is impossible to say whence the 
 philosophers of the thirteenth century could have got any thing like 
 a set of rules. They certainly had not got any thing Greek, except 
 from the Arabs: the system of Boethius does not give rules of 
 computation. Moreover, the name algorithm was in later times so 
 invariably connected with Arabic arithmetic, that the presumption 
 is strong it was so from the beginning. Now Roger Bacon, the con- 
 temporary of Sacrobosco, not only used the adjective algoristicus 
 several times, but recapitulates the names ofa set of rules. The 
 theologian, he says (Jebb, p. 138), should abound in the power of 
 numbering, that he may know all the algoristic modes, not only for 
 integers, but for fractions, to numerate, to add, to subtract, to me- 
 diate (divide by two), to multiply, to divide, and to extract roots. 
 Now this is precisely the set of rules given in the treatise attributed 
 to Sacrobosco, with one omission: this last treatise distinguishes 
 duplation (multiplication by two) from other cases of multiplication, 
 which Bacon does not; and progression, which is, however, only a 
 mixture of other rules. 
 
 There is a sentence of Bacon’s contemporary, Matthew Paris, 
 quoted by Mr. Hallam, in which he speaks of what can be done 
 
WORKS ON ARITHMETIC. 15 
 
 with Greek notation, as being what cannot be done either in Latin, 
 or in algorithm, vel in algorismo. These words might easily be 
 interpolated; so instead of using them, as Mr. Hallam does, to 
 give ground of presumption that Bacon had the Arabic notation, I 
 should rather use what I have drawn from Bacon to strengthen 
 the genuineness of the words of Paris; and I think Mr. Hallam will 
 be ready to do'the same. 
 
 In ‘Rara Mathematica,’ London, eighteen-forty-one, octavo, 
 second edition, Mr. Halliwell has inserted this treatise of Sacrobosco, 
 together with the poem de algorismo attributed to him in some of 
 the manuscripts, but probably enough without foundation. The 
 seven rules above mentioned, are thus given : 
 
 Septem sunt partes, non plures, istius artis; 
 Addere, subtrahere, duplareque dimidiare 
 
 Sextaque dividere est, sed quinta est multiplicare 
 Radicem extrahere pars septima dicitur esse: 
 
 on which Mr. Halliwell quotes a manuscript of perhaps nearly as 
 early a date as Sacrobosco, which describes the seven rules and the 
 necessity of the letters called figures, as follows: 
 
 En argorismo devon prendre Et de radix enstracion 
 
 Wil especes .-. iy A chez vii especes savoir 
 Adision Subtracion Doit chascvn en memoire avoir 
 Doubloison mediacion Letres qui figures sont dites 
 Monteploie et division Et qui excellens sont ecrites 
 
 In the same ‘Rara Mathematica’ are to be found an English 
 manuscript of the 14th century ‘On the numeration of Algorism,’ 
 and John Norfolk ‘In artem progressionis summula,’ dated four- 
 teen-forty-five at the end. 
 
 Paris, fifteen-twenty-six. John Martin Siliceus. 
 ‘ Arithmetica Joh. Martini Silicei theoricen praxinque luculen- 
 ter complexa’ &c. Folio in threes. 
 
 This is a third edition by Thomas Rhetus; Heilbronner says 
 that the first was Paris, fifteen-fourteen, and that the author, who was 
 a professor at Salamanca, named Gujieno (Silex), died at Toledo 
 in fifteen-fifty-seven. Montucla says he was Archbishop of Toledo. 
 This is a work of considerable extent of subject, and seems to have 
 no great fault except the usual one of prolixity. It gives a treatise 
 on theoretical arithmetic (Boethian in character), one on the rules 
 of computation, one on the mode of calculating by pebbles or the 
 abacus, and one on fractions. The second edition, fifteen-nineteen, 
 by Orontius Finzus, is in the Royal Society’s Library. 
 
 Paris, fifteen-twenty-eight. John Fernel. ‘De pro- 
 portionibus Libri duo.’ Folio in threes. 
 A book on proportion of this date is in great part filled with 
 
 Boethian arithmetic. This book deserves attention : its author had 
 a much better grasp of Euclid than most of his contemporaries. 
 
16 A CHRONOLOGICAL LIST OF 
 
 Leyden, fifteen-thirty-one. Joach. Fortius Ringel- 
 bergius.- ‘De Ratione Studi.’ Octavo (duodecimo size). 
 
 At the end, same place, year, and form, with another title-page, 
 apparently another work, but of this I am not sure, is J. F. R. 
 ‘Compendium de scribendis versibus,’ to which is attached ‘ Arith- 
 metica’ (O), the most Lilliputian treatise I know of, giving in 17 
 pages an epitome of Boethian terms, algorithmic rules, and counter 
 calculation. [It is said that the author’s name was Sterk. | 
 
 Venice, Giovanni Sfortunati. ‘Nuovo Lume, 
 Libro di Arithmetica . . . . Composto per lo acutissimo pre- 
 scrutatore delle Archimediane, et Euclidiane dottrine Giovanni 
 Sfortunati da Siena.’ Quarto in fours. 
 
 The end of the work torn out, so I cannot give the date. He 
 mentions L. de Burgo, Calandri, and P. Borgio, and Tartaglia men- 
 tions him: which gives certain limits. By the manner in- which the 
 author is described, this must be a reprint: it is not likely a work 
 would be originally printed at Venice, one cause of the production 
 of which is stated to be the barbarism of the Venetian dialect. I 
 can find no mention of Sfortunati in catalogues. Cardan, writing 
 in fifteen-thirty-seven, mentions the work of Fortunatus, the same 
 as the above, no doubt. P. 458, 460. 
 
 Nuremberg, fifteen-thirty-four. ‘ Algorithmus demonstra- 
 tus.’ Quarto. 
 
 John Schoner, the editor, attributes this to Regiomontanus. It 
 is a series of demonstrated propositions of numbers connected with 
 the Arabic notation, involving not only the rules of computation, 
 but such as that the number of figures in the cube cannot ex- 
 ceed three times that in the root, &c. J. Schoner, who edited several 
 writings of Regiomontanus, is likely enough to be right on this one, 
 which is moreover not unworthy of such an author. Itis not in the 
 list which he published himself, but it seems to be generally recog- 
 nised. (Weidler zn verb.) 
 
 Paris, fifteen-thirty-five. Orontius Fineus. ‘ Arith- 
 metica practica libris quatuor absoluta..... Recens ab Au- 
 thore castigata. .... hoe est in nativum splendorem (quem 
 priorum impressorum amiserat incuria) summa fidelitate res- 
 tituta.’ Folio in fours. 
 
 This must be at least the third edition ; Leslie says the first was 
 in fifteen-twenty-five. Orontius died in 1555. 
 
 Strasburg, fifteen-thirty-six. Hludalrich or Huldrich 
 Regius. ‘ Utriusque Arithmetices Epitome ex variis autho- 
 ribus concinnata per Hudalrichum Regium.’ Octavo (duode- 
 cimo size). 
 
 The first book is on the Boethian arithmetic: the second on the 
 
‘WORKS ON ARITHMETIC. 17 
 
 rules of computation with integers and fractions, and on the use of 
 the abacus. There are but 96 small folios with large print: so that 
 if I wished to bring any person into the closest contact with the 
 middle ages at the least expense of reading, with reference both 
 to their mode of expression and operation, I should certainly pre- 
 scribe this book. 
 
 Paris, fifteen-thirty-eight. IWicomachus of Gerasa. ‘ Ni- 
 xomarvov Legaowou Agibunrinns BiPAse dvo Nicomachi Gerasini 
 Arithmeticee Libri duo. Nune primum typis excusi, in lucem 
 eduntur. Parisiis in officina Christiani Wecheli.’ Quarto. 
 
 Nicomachus, who lived about the time of Tiberius, was a Pytha- 
 gorean, and of course devoted to arithmetic. He was in his day, 
 as Lucian intimates, a Cocker, a person whose name passed as an 
 allusion to reckoning and numbering. The work of Boethius is 
 wholly drawn from him: some people call it a. mere translation 
 of, others a comment on, Nicomachus. Perhaps a free rendering 
 might be correct. This edition is Greek without Latin: [there is 
 another, also Greek without Latin, at the end of the Zheologumena 
 Arithmetice attributed to Jamblichus, Lezpsic, eighteen-seventeen, 
 octavo (?)]. It is then ultimately to Nicomachus being a Pytha- 
 gorean that we owe those once never-ending denominations of 
 numbers and ratios, of which it might now be difficult to trace any 
 vestige in modern language, except the common words multiple and 
 sub-multiple and the sesquialter stop of an organ. 
 
 Milan, fifteen-thirty-nine. Jerome Cardan. ‘Practica 
 Arithmetice et mensurandi singularis.’ Octavo (QO). 
 
 This was reprinted, under the title of ‘ Practica Arithmetica,’ 
 in the fourth volume of his collected works, Leyden, sixteen-sixty- 
 three, ten volumes, folio in twos. On the arithmetic there is no re- 
 mark to make, except that, as might be expected from an Italian of 
 that day, Cardan shews more power of computation than the French 
 and German writers. There is a chapter recapitulating the num- 
 bers which have mystic properties as he calls them, one use of which 
 is in foretelling future events. These are mostly the numbers men- 
 tioned in the Old and New Testaments, but not altogether : 400, for 
 instance, makes its appearance, because it was (but it was not) the 
 number of bishops at the Nicene Council. In the tenth volume of 
 the same collection is an unfinished treatise by Cardan, headed 
 ‘Artis Arithmetic Tractatus de Integris,’ which seems to be the 
 commencement of a more extensive treatise than the former one. 
 It may be cited from it that decidedly, in Cardan’s opinion, it was 
 Leonard of Pisa who first introduced the Arabic numerals into 
 Europe. 
 
 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. + 
 
 c 2 
 
 LIBRARY 
 UNIVERSITY OF ILLINOIS 
 
18 A CHRONOLOGICAL LIST OF 
 
 Newlye invented and compyled by Syr Rycharde Benese, 
 Chanon of Marton Abbay besyde London. Prynted in South- 
 warke 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. { Im- 
 pynted at London, by Thomas Colwell.’ 
 
 The approximate date of the first we must guess at from the fact 
 that James Nicolson’s dated works are from fifteen-thirty-six to 
 thirty-eight, and from the presumption that a printer’s undated works 
 come before the dated ones. Thomas Colwell’s dated works are from 
 fifteen-fifty-eight to seventy-five. The acre is four roods, each rood 
 is ten daye-workes, each daye-worke four perches. So the acre being 
 40 daye-workes of 4 perches each, and the mark 40 groats of 4 pence 
 each, the aristocracy of money and that of land understood each 
 other easily.* The reprint of the above work wants a set of tables 
 which were in the original, and which I take to be the earliest 
 mathematical tables published in England, except the smaller tables 
 of the same kind which appeared in some acts of parliament. They 
 are of double entry, the entries being made by perches in length 
 
 and breadth. Thus under 79 opposite 9 we find a meaning that 
 
 79 perches by 9 is 4 acres 1 rood, 7 day-works and 3 square perches. 
 Perhaps this sort of table became common: I find it, without the 
 slightest deviation of form, in Arthur Hopton’s ‘ Baculum Geodeti- 
 cum sive Viaticum, or the Geodeticall Staffe,’ London, sixteen-ten, 
 quarto. 
 
 it 
 
 Paris, fifteen-thirty-nine. Johannes Noviomagus. ‘ De 
 Numeris libri duo.’ Octavo (small) (0). 
 
 Professedly on computation, but interspersed with many curious 
 historical remarks. It contains the first hints I have found of the 
 most probable explanation of the origin of the Roman numerals. 
 
 Paris, fifteen-forty-four. Orontius Finzeus. ‘< Arith- 
 metica Practica . . . . . multisque accessionibus locupletata.’ 
 
 Octavo (QO). 
 
 * More easily than they do at the time when this is written. In the old day, when 
 the aristocracy of money was a Jew, the aristocracy of land used to.shut him up, and 
 draw a tooth a day until he protected agriculture to the amount demanded. When 
 Christians became wealthy, a tax upon the use of the teeth seems to have been sub- 
 stituted for their forcible removal. The debates on the abolition of this tax are now 
 proceeding. But, speaking of weights and measures, there is something else which 
 tells a tale about the feasts and Sundays of old England. The sack of wool was 13 
 tods of 28 pounds each, or 364 pounds. This was arranged, that the tasks of those who 
 spun, &c. might be easily calculated; a pound a day being a tod a month and a sack 
 a year. But where are the Sundays and holidays? Most likely they were made up 
 oun the other days. 
 
WORKS ON ARITHMETIC. 19 
 
 Paris, fifteen-fifty-five. Orontius Finzeus. ‘De Arith- 
 metica Practica libri quatuor.’ Quarto (QO). 
 
 An augmented edition. Had this book been simply headed as 
 arithmetic, I might have let it alone. But as it is called prac- 
 tical arithmetic, it must stand as an index of the great advance of 
 the English over their neighbours in computation, as shewn by Ton- 
 stall and Recorde. P. 428, 436. 
 
 Nuremberg, fifteen-forty-four. Michael Stifel. <‘ Arith- 
 metica integra. Authore Michaele Stifelio. Cum preefatione 
 Philippi Melancthonis.’ Quarto. (Hutton’s Tracts, v. 11. p. 237.) 
 
 The two first books are on the properties of numbers, (O), on surds 
 and incommensurables, learnedly treated, and with a full knowledge 
 of what Euclid had done on the subject. The third book is on alge- 
 bra, and passes for the introduction of algebra into Germany. But 
 Stifel himself, in his preface, acknowledges his obligations to Adam 
 Risen, and professes to have taken all his examples from Christopher 
 Rudolph. Of the latter I can find nothing except a statement 
 that his work was subsequently published: but Kastner (v. i. p. 108) 
 gives some information on the works of Adam and Isaac Riese. 
 P. 424, 436, 474, 450. 
 
 Stifel is supposed to have been the inventor of the signs + and 
 — to denote addition and subtraction : and the manner in which he 
 speaks of these signs seems to imply that they were of his own 
 introduction (p. 110). It is “ we place this sign,” &c. and ‘‘ we say 
 that the addition is thus completed;”’ and so on. The sign +, in 
 the hands of Stifel’s printer, has the vertical bar much shorter than 
 the other : and when it is introduced in the woodcuts by the engraver, 
 the disproportion is greater still. One would (supposing the wood- 
 engraver to have imitated the handwriting) have supposed that — 
 was first used, and that + was derived from it by putting a small 
 eross-bar for a distinction. This form is rather against the late 
 learned Professor Rigaud’s idea (see Mr. Davies’ Solutions of Hutton’s 
 Mathematics, p. 11) that + is a corruption of P, the initial of plus, 
 and also against Mr. Davies’ ingenious conjecture that it is a cor- 
 ruption of et or & Stifel does not call the signs plus and minus, 
 but signum additorum and signum subtractorum. In no instance do 
 I find him using the first pair of appellatives at all. 
 
 I had written the above, and become fully impressed, by repeated 
 examination, with the suspicion that the vertical bar in + was, in 
 Stifel’s invention, a small distinction superadded on the sign —, when 
 I happened, in the immediate preparation of these sheets for press, 
 to remember Colebrooke’s statement that the Hindoos (our original 
 teachers in algebra) use a dot for subtraction, and nothing but the 
 absence of the dot for addition. It may be likely then, that in the 
 first instance the Hindoo dot was elongated into a bar, to signify 
 subtraction, addition having no sign: and that the first who found 
 it convenient to introduce a sign for addition, merely adopted the 
 sign for subtraction with a difference. 
 
20 A’ CHRONOLOGICAL LIST OF 
 
 Against the above conjecture it may be argued, that the European 
 algebra comes from India through the Mahometan writers, who use 
 no signs at all. On this I can only remark at present, that I have 
 long suspected, from many circumstances, that there was a more 
 direct communication with India in the introduction of algebra than 
 any one now believes. 
 
 Dr. Ritchie suggested that perhaps + was used to denote addi- 
 tion as being two marks joined together, and made significative of 
 two numbers joined together: and that — denoted subtraction, as 
 indicating what is left after one of the marks of the junction is 
 removed. This suggestion is more like what ought to have been 
 the derivation than any I have seen: but there is little reason to 
 suppose that any such attempt at significant symbols would have 
 been made in the sixteenth century. 
 
 M. Libri attributes the invention of + and — to Leonardo Da 
 Vinci, in whose manuscripts he says he had seenthem. But as it 
 happened to me to discover, in a manuscript of that justly celebrated 
 man (if it be not more correct to say unjustly celebrated, as not cele- 
 brated enough for his merits) now in the British Museum, the sign 
 + used for the figure 4, I think the question must rest open until 
 citations which establish the point without mistake can be produced. 
 
 No place, nor date. Wm. Buckley. ‘ Arithmetica me- 
 morativa sive compendaria Arithmeticee tractatio.’ Apparently 
 octavo. 
 
 This book has been described both by Leslie and Peacock, who 
 have severally quoted some of the verses of which it consists. They 
 say it was first printed in fifteen-fifty, and afterwards at the end of 
 Seton’s Logic in sixteen-thirty-one. I should probably not have seen 
 it to this day (for after Leslie’s full citations, I should hardly have 
 made active search forit), had not an anonymous correspondent (whom 
 I have only this way of thanking) obligingly forwarded to me an im- 
 perfect copy which he happened to find among his papers. It is, I 
 suppose, the extract from Seton’s Logic: the signature in the title- 
 page is Q 2, after a /ibellus ad lectorem, ‘ Quisquis Arithmeticam . .. . 
 qui meminisse dedit :’ and there is a preface signed T. H. It is 
 worth noting that Buckley’s Latin name, Bucleus, has been fre- 
 quently written Budeus by the continental bibliographers, so that 
 in all probability the learned author of the book De Asse has been 
 taken for the author of these verses. Hylles callshim Buckle. P. 437. 
 
 [ According to Watt, there were editions of Seton’s Logic, with 
 ‘Huic accessit ob artium ingenuarum inter se cognationem Arith- 
 metica Gulielmi Buclei,’ London, fifteen-seventy-two, seventy-four, 
 and seventy-seven, octavo. | 
 
 Basle, fifteen-fifty. John Scheubel. ‘Euclidis Mega- 
 rensis..... Algebree porro Regulze .. . . . his libris preemissee 
 sunt... Folio in twos. | 
 
 The algebra was reprinted, Paris, fifteen-fifty-two. Quarto. ‘ Al- 
 
WORKS ON ARITHMETIC. 21 
 
 gebree compendiosa facilisque descriptio,’ &c. For a description of 
 the work see Hutton’s Tracts, vol. ii. p. 241. 
 
 Leyden (Lugduni, without further description), fifteen-fifty. 
 Henry Cornelius Agrippa. ‘De Occulta Philosophia 
 Libri Tres.’ Octavo (Italic letter). 
 
 As the preface is dated Mechlin, fifteen-thirty-one, and [an edi- 
 tion appeared there in fifteen-thirty-three], this latter must be the 
 first. ‘This book has as much right in my list as that of Jordano 
 Bruno presently noticed, treating ofnumbers in much the same way. 
 It isa property ofseven that there are seven angels who stand before 
 God; of twelve, that there are twelve apostles, twelve permutations 
 of the letters of the tetragrammaton, &c. A good many of the more 
 formal works on arithmetic of this period abound in enumerations of 
 the same kind, evidently considered as appertaining to the higher 
 uses of the science. 
 
 Augsburgh, fifteen-fifty-four. Psellus. ‘De quatuor 
 scientiis mathematicis.’ Edited by Xylander. Gr. Lat. Oc- 
 tavo (duodecimo size). 
 
 Psellus lived in the ninth century. The part relating to Arith- 
 metic is a mere compendium of arithmetical terms, perhaps from 
 Boethius. 
 
 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 (Q). 
 
 Of this enormous book I may say, as of that of Pacioli, that it 
 wants a volume to describe it. P. 430,450. It was partly trans- 
 lated into French as follows: Paris, sixteen-thirteen. ‘ L’Arith- 
 metique de Nicolas Tartaglia, Brescian, grand mathematicien et 
 prince de praticiens, recueillie et traduite d’Italien en Francois par 
 Guillaume Gosselin de Caen.’ Octavo. 
 
 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. 
 
 On this see the Companion to the Almanac for 1837, p. 36; Hut- 
 ton’s Tracts, v. ii. p. 248; P. 369. It is rarely remembered that the 
 old name of Algebra, the cossic art (from cosa, thing), gave this first 
 English work on algebra its punning title the whetstone of wit, Cos 
 ingenii. 
 
 Lione, fifteen-fifty-eight. Giov. Franc. Peverone. ‘Due 
 
22 A CHRONOLOGICAL LIST OF 
 
 brevi e facili trattati. Il primo d’Arithmetica: l’altro di Geo- 
 metria, &c. Quarto. 
 
 A book of the very simplest examples of computation. 
 
 Leyden, fifteen-fifty-nine. Joh. Buteo. ‘Joan. Buteonis 
 Logistica quee et Arithmetica vulgo dicitur.’ Quarto in fours. 
 (0). 
 
 In the same year was published, by the same author, his Opera 
 Geometrica, the first of which is a plan of Noah’s ark, with calcula- 
 
 tions as to its sufficiency for holding all the animals and their pro- 
 vender. P. 437. 
 
 Paris, fifteen-sixty. James Pelletier. ‘Jacobi Pele- 
 tarii Cenomani, de occulta parte numerorum, quam algebram 
 vocant, libri duo.” Quarto. [First edition, fifty-eight. | 
 
 This is said to be the first French work on algebra. There is 
 little purely arithmetical in it: at the end is a table of squares and 
 cubes up to those of 140. See Hutton’s Tracts, vol. ii. p. 245. 
 
 London, fifteen-sixty-one. Robert Recorde. ‘The 
 Grounde of artes: Teaching the worke and practise of Arith- 
 metike, 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 overseen and augmented with new and necessarie 
 Additions. 
 
 I{ohn] D[ee}. 
 All youth and Elde that reasons Lore 
 Within your breastes will plant to trade 
 
 Of Numbers might the endles store 
 Fyrst vnderstand, than farther wade’ 
 
 Octavo (duodecimo size) (O). 
 
 See the Companion to the Almanac for 1837, p. 32. That this 
 - book was published about fifteen-forty there is internal evidence: and 
 Tanner gives it that date. Dr. Peacock has fifteen-forty-two. But I 
 have never seen any edition earlier than this. John Dee published 
 it again, London, fifteen-seventy-three, octavo, with other verses in 
 the title-page, and the original dedication to Edward VI. restored. 
 The preceding edition, though published under Elizabeth, was pro- 
 bably arranged in the life of her sister, which may account for the 
 omission of the dedication. There have been many editions. Hart- 
 well’s edition of Mellis’s edition was published thrice at least, in 
 sixteen-forty-eight and sixteen-fifty-four, and once more with the 
 date torn out in our copy. Mellis added a third part, on practice 
 and other things: in Roman letter, the sacred text of Recorde hav- 
 ing always its old black letter. The last edition I know of it is by 
 Edw. Hatton, London, sixteen-ninety-nine, quarto; it has an addi- 
 
WORKS ON ARITHMETIC, 23 
 
 tional book called ‘Decimals made easie.’ P. 408-10, 434, 454, 
 476. 
 
 Vienna, fifteen-sixty-one. Christoffer Rudolff. ‘ Kunst- 
 liche rechnung mit der ziffer und mit der zalpfenningen,’ &c. 
 Octavo (small). 
 
 This is the man to whom Stifel refers as his instructor in algebra. 
 But there is nothing like either of the signs + or — in his book: so 
 that Stifel does not appear to have got them from him. This book 
 is not algebra: and whether any thing of Rudolph’s deserving of 
 that name was published is more than I can now settle. 
 
 Antwerp, fifteen-sixty-five. Walentine Menher de 
 Kempten. ‘Practicque pour brievement apprendre a Cif- 
 frer, et tenir livre de Comptes, avec la regle de Coss, et Geo- 
 metrie.’ Octavo. [Date at the end, sixty-four, July. | 
 
 Here are books on Arithmetic, Book-keeping, Algebra, Geometry, 
 and Trigonometry : it appears that part of the arithmetic had been 
 published in fifteen-fifty-six, of the geometry in fifteen-sixty-three, of 
 the algebra in fifteen-fifty-six. The arithmetic is an early specimen 
 of the book of mererules. In this and other French books of the same 
 date, and even much later, the words septante and nonante are used 
 for seventy and ninety. The book-keeping is a short treatise, or 
 rather example, of double entry. The algebra is more extensive, 
 on the model of Stifelius, with many questions leading to simple 
 equations, each embellished with an illustrative picture in wood. 
 The geometry is mensuration. The spherical trigonometry is from 
 Regiomontanus, and refers to no other tables. This book has some 
 pretension to be called a course of mathematics. Menher is often 
 quoted in England for his tables of rebate or discount. 
 
 Antwerp, fifteen-sixty-seven. Pedro Nunez. ‘ Librode 
 algebra en arithmetica y geometria.’ Octavo (small). 
 
 This book is wholly algebraical. See Hutton’s Tracts, vol. ii. 
 p. 250. : 
 
 London, fifteen-sixty-eight. Hftumfrey Baker. ‘The 
 Well Spryng of Sciences, which teacheth the perfecte woorke 
 and practise of Arithmeticke . . . . beautified with moste ne- 
 cessary rules and questions . > Octavo (very small size) 
 (O). 
 
 The reprints will be presently mentioned. 
 
 Paris, fifteen-seventy-one. Alex. Wandenbussche. 
 ‘ Arithmetique militaire.’ Quarto. 
 
 A short work, mostly on the arithmetic of the arrangement of 
 troops, with a large number of military maxims and observations. 
 
24 A CHRONOLOGICAL LIST OF 
 
 The first book ends with a question described as non moins plaisante 
 que belle, as follows. A soldat désespéré asked of an arithméticien 
 fantastique how far it was from Milan to hell. The mathematician, 
 placing it at the centre of the earth, made the distance 11454 leagues. 
 This, and Je prie a Dieu nous garder d’y aller, is the whole of the joke. 
 
 London, fifteen-seventy-one. Leonard Digges. ‘A Geo- 
 metricall Practise, named Pantometria . . . . framed by Leo- 
 nard 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 Geo- 
 metrical practical treatize named Pantometria. ... . First pub- 
 lished by Thomas Digges Esquire... . lately reviewed by the 
 author himself,’ &c. Quarto. 
 
 Almost the whole is application of Arithmetic to Geometry, in 
 measurement both of planes and solids. 
 
 In this book I find the earliest printed mention I ever met with 
 of the theodelite : a word the derivation of which has long been a 
 puzzle. It is known to be an exclusively English word in all time 
 preceding the middle of the last century. Digges calls it ‘ the circle 
 called theodelitus.’ A moveable radius travelling round a circle, for 
 the measurement of angles, had been long denoted by the Arabic 
 word alhidada (whence the French word alidade). Theodelite seems 
 unlikely to be a corruption of this, at first: nor should I have sus- 
 pected such a thing, if 1 had not found in Bourne’s ‘Treasure for 
 travailers,’ London, fifteen-seventy-eight, guarto, the intermediate 
 formation, athelida, used for the same thing. I hold it then pretty 
 certain that this is the true origin of the word theodelite, which 
 was so spelt, not theodolite. (See the Philosophical Magazine, April, 
 1846.) 
 
 Venice, fifteen-seventy-five. Franc. Maurolycus. ‘Arith- 
 meticorum libri duo.’ Quarto in fours. 
 
 On the properties of numbers and the doctrine of incommen- 
 surables; a superior work to the mass of those which then treated 
 of similar subjects. 
 
 Basle, fifteen-seventy-nine. Christian Urstisius. ‘Ele- 
 menta Arithmeticee, logicis legibus deducta.’ Octavo (duo- 
 decimo size) (O). 
 
 It is said that Urstisius was a follower of Copernicus and a 
 teacher of Galileo. This Arithmetic was translated by T. Hood, 
 London, fifteen-ninety-six, octavo (small). ‘The Elements of Arith- 
 meticke most methodically delivered,’ &c. 
 
 Brescia, fifteen-eighty-one. Ant. Maria Vicecomes 
 (Visconti). ‘ Practica numerorum et mensurarum,’ &c. Quarto. 
 
WORKS ON ARITHMETIC. 25 
 
 A book of mixed arithmetic and algebra, in which the modern 
 method of division is used. It would, I think, repay an examina- 
 
 tion, particularly on the part which relates to. the extraction of 
 roots. 
 
 Antwerp, fifteen-eighty-one. Gemma Frisius. ‘ Arith- 
 meticee practicee methodus facilis, in eandem J. Stein et J. 
 Peletarii annotationes,’ &c. Octavo (small size) (QO). 
 
 This is a reprint, much augmented, of a compendious work of 
 Frisius, the ‘Ar. pract. meth. fac.’ Witteberg, fifteen-sixty-three, 
 octavo (small). But there is an earlier edition of Gemma Frisius, 
 also by Peletarius, being the earliest I have met with, as follows : 
 
 Paris, fifteen-sixty-one. ‘ Arithmeticee Practicee 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 additze sunt Radicis utriusque demonstra- 
 tiones.’ Octavo (small) (QO). 
 
 These demonstrations of the rules for the square and cube roots 
 Peletarius treats as a great boon to his reader: and states that he 
 was very near contenting himself with a reference to his book on 
 occult properties of numbers (his Algebra of 1560, before mentioned). 
 
 London, fifteen-eighty-three.e Hl. 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). 
 
 Reprinted (same place and size) in sixteen-fifty-five, with a re- 
 ference at the end to an edition of sixteen-forty-six. It is one of 
 the books which break the fall from the ‘ grounde of Artes’ to the 
 commercial arithmetics of the next century. There are some short 
 rules for particular cases, and great attention to the rule of practice. 
 Among the peculiarities of the book is a notion, apparently, that none 
 but fractions should deal with fractions: for Baker will not double 
 FD A instance, by multiplying by 2, but only by dividing by 3. P. 
 
 London, fifteen-eighty-four. John Blagrave. ‘The ma- 
 thematical jewel.’ Folio in twos. 
 
 See the Companion to the Almanac for 1837, p. 41. This book 
 is curious, because the woodcuts are all done by the author’s own 
 hand, and Walter Venge, the printer, is not known to have printed 
 any other work. [For Blagrave’s legacy tothe three parishes of 
 Reading, see Ashmole’s Berkshire, y. iil. p. 372; for the appear- 
 
 D 
 
26 A CHRONOLOGICAL LIST OF 
 
 ance of his ghost, ‘credibly reported by many honest and discreet 
 persons,’’ see the Annus Mirabilis, 1662, p. 49.] This book is not 
 arithmetical, but the contrary: it is an avowed attempt to drive 
 computation out of astronomy by the introduction of an instrument 
 called the Jewel, which is a projection of the sphere. Of the tables 
 ef sines he says, ‘ But now there are in like maner an infinite num- 
 ber of intricate questions more harder far than any yet propounded, 
 which by Regiomont. Copernicus and others doctrine, grow to great 
 toile with their Synes, calculations, and proportions, wherein first 
 they hunt about for one Syne, which they call inventum primum, 
 then for another, which they call inventum secundum, and commonly 
 mventum tertium, and perhaps guartum, and so foorth. After all these 
 are found, then they multiplie and devide them, and compare their 
 proportions: and when al is done, that they have founde the Seyne 
 sought for: they yet are faine to goe to their tables for the arch cor- 
 respondent.’ ‘There was a great disposition at this time among the 
 English to try to substitute instruments for computation. Blagrave’s 
 Jewel stood in high estimation ; and as late as 1658, John Palmer, 
 who published a description of it under the name of the Catholic 
 Planisphere, says it was a subject of frequent inquiry where the in- 
 strument and the description were to be met with. 
 
 Leyden, fifteen-eighty-five. Simon Stevinus. ‘LL’ Arith- 
 metique de Simon Stevin de Bruges: Contenant les computa- 
 tions 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 mainte- 
 nant premierement traduicts in Francois. Encore un livre 
 particulier de la Pratique d’Arithmetique, contenant entre 
 autres, Les Tables d’Interest, La Disme; Et un traicté des In- 
 commensurables grandeurs: Avec lExplication du Dixiesme 
 livre d’Euclide.’ Octavo. 
 
 Leyden, sixteen-thirty-four. Albert Girard. ‘Les 
 Oeuvres Mathematiques de Simon Stevin de Bruges,’ &c. 
 Folio in threes. 
 
 See Hutton, Tracts, vol. ii. p. 257, and Peacock, Enc. Metr. 
 Arithmetic, p. 440. My copy of the first work is imperfect, stopping 
 at the end of the Diophantus; but as Alb. Girard gives every thing 
 in the collected works, just in the order announced in the above title- 
 page, I suppose there can be no doubt that every thing there: an- 
 nounced was published in 1585. The Disme is the first announce- 
 ment of the use of decimal fractions. Dr. Peacock supposes it was 
 first published in Dutch about 1590: we here see that it was pub- 
 lished in French in 1585. Hutton speaks of a Dutch edition in 
 1605. I do not know where either date* was got from. Qn look- 
 ing at the verses with which Stevinus’s friends (according to .cus- 
 
 * Perhaps 1605 is a mistake for 1626, at which cate [R. Soc. Libr. Cat.] De Thiende, 
 Jecerende alle rekeninghen,.&c. was published at Gouda. 
 
WORKS ON ARITHMETIC. 27 
 
 tom) have eulogised him, a part of a book to which I am now and 
 then under obligation, I find that the Disme, &c., was attached to 
 the edition of 1585, with a new paging. Jagius Tornus, philomathes, 
 be he whom he may, quotes and writes as follows, with a side-note: 
 
 Non fumum ex fulgore sed ex fumo dare lucem 
 
 Cogitat, ut speciosa dehine miracula promat. 
 
 Sume unum é multis. quid non Decarithmia prestat Decarithmia 
 Divinum scriptoris opus? cui non ego si vel la ee 
 Aurea mi vox sit, centum lingue, ordque centum ager ce 
 Omni etate queam laudes persvulvere dignas. 
 
 Accordingly, assuming Girard to have faithfully copied his original 
 in his headings (which he has done as far as my means of com- 
 parison go), the method of decimal fractions was announced before 
 1585, in Dutch. 
 
 The characteristic of Stevinus is originality, accompanied by 
 a great want of the respect for authority which prevailed in his 
 time. For example, great names had made the point in geometry 
 to correspond with the unit in arithmetic: Stevinus tells them that 0, 
 and not 1, is the representative of the point. And those who cannot 
 see this, he adds, may the Author of nature have pity upon their 
 unfortunate eyes; for the fault is not in the thing, but in the sight 
 which we are not able to give them. P. 426, 440, 460. 
 
 [Dr. Peacock (p. 440) mentions an English translation of this 
 tract by Richard Norton, sixteen-eight, under the title ‘ Disme, the 
 arte of tenths, or decimal Arithmetike..... invented by the excel- 
 lent mathematician, Simon Stevin.’] . 
 
 Since writing the above I have examined the account of Stevinus 
 given by M. Quetelet, presently referred to. It is stated that the 
 first work of Stevinus was his table of interest, Antwerp, fifteen- 
 eighty-four ——; this was reprinted the next year in the Arithmetic. 
 In the Supplement of the Penny Cyclopedia, ‘Tables,’ I have noted, 
 from the contents of this table and ofthe Disme, the great proba- 
 bility that the table of compound interest suggested decimal fractions : 
 the only doubt was, which was written first, the table or the Disme. 
 The statement of M. Quetelet clears this up: and I hold it now next 
 to certain that the same convenience which has always dictated the 
 decimal form for tables of compound interest was the origin of deci- 
 mal fractions themselves. 
 
 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 Me- 
 moriall Journall and Leager, and of other necessaries apper- 
 taining 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 forth by John Mellis Scholemaister, 1588. 
 
28 A CHRONOLOGICAL LIST OF 
 
 Imprinted at London by John Windet, dwelling at the signe 
 of the white Beare, nigh Baynards Castle. 1588.’ Octavo. 
 
 This is the earliest English book on book-keeping by double 
 entry which has ever been produced. At the end of the book-keep- 
 ing is a short treatise on arithmetic (O). But Mellis says: ‘ Truely 
 Iam but the renuer and reviver of an auncient old copie printed 
 here in London the 14 of August, 1543. Then collected, pub- 
 lished, made and set forth by one Hugh Oldcastle Scholemaster, 
 who, as appeareth by his treatise then taught Arithmetike and this 
 booke, in Saint Ollaves parish in Marke Lane.’ 
 
 [ Watt and others (but not Ames) mention a quarto treatise having 
 this date, fifteen-forty-three, and entitled ‘ A profitable Treatyce 
 called the Instrument or Boke to learne to knowe the good order of 
 the kepyng of the famouse reconynge, called in Latyn, Dare and 
 Habere, and in Englyshe, Debitor and Creditor.’ But no author’s 
 name is given. } 
 
 Beckman quotes Anderson as to a book of James Peele, London, 
 fifteen-sixty-nine, folio, on book-keeping: but it may be doubted 
 whether this was not a book on single entry. 
 
 London, fifteen-ninety. Cyprian Lucar. ‘A treatise 
 named Lucarsolace,’ &e. Quarto. 
 
 A work on mensuration, interspersed with arithmetical rules. It 
 contains the description ofa fire-engine. We find from it that the 
 surveyors used, not black-lead, but ‘fine-pointed coles or keelers,’ 
 of which three or four might be bought of any painter for a penny. 
 Lucar also translated part of Tartaglia on Artillery, with a long 
 appendix. London, fifteen-eighty-eight. Solio in threes. 
 
 London, fifteen-ninety. Thomas Digges. ‘An Arith- 
 metical warlike treatise named Stratioticos. . . . first published 
 by Thomas Digges Esquire, anno Salutis, 1579,’ &c. Quarto (QO). 
 
 There is here a brief and good treatise on arithmetic, and some 
 
 algebra of the school of Recorde and Scheubel: but the greater part 
 of the work is on military matters. 
 
 Helmstadt, fifteen-ninety. WHeizo Buscher. ‘ Arith- 
 meticee libri duo.” Octavo (small) (Q). 
 
 A short and very backward work. 
 
 Bergomi, fifteen-ninety-one. Peter Bungus. ‘ Numero- 
 rum Mysteria.’ Second edition. Quarto in fours. 
 
 Dr. Peacock gives some description of this fantastical work, 
 page 424. 
 
 Frankfort, fifteen-ninety-one. Jordano Bruno. ‘De 
 Monade numero et Figura liber Consequens Quinque de Min- 
 
WORKS ON ARITHMETIC, 29 
 
 imo magno et Mensura. Item de Innumerabilibus, immenso, et 
 Infigurabili; seu De Universo et Mundis libri octo.’? Octavo 
 (duodecimo size). 
 
 On Jordano Bruno see Bayle, Drinkwater-Bethune’s Life of 
 Galileo, p. 8, and Libri, vol. iv. p. 141, &c. who makes him a vorticist 
 before Des Cartes, an optimist before Leibnitz, a Copernican before 
 Galileo. The end of it was that he was roasted alive at Rome, 
 February 17, 1600, at the age of fifty: ostensibly for heresy (he 
 had been a Dominican), but most probably in revenge for his sa- 
 tirical writings. He does not seem, however, to have been a Pro- 
 testant martyr; as his opinions would probably have procured his 
 death in almost any state in Europe, as matters were at that time. 
 Bruno, says M. Libri, seems to have become a Copernican by a sort 
 of intuition, for he was any thing rather than a mathematician. His 
 book on monad and number fully confirms this last statement: it is 
 a collection of dissertations on individual numbers. But it has the 
 advantage over Pacioli and Bungus (P. 424), that the Latin is better, 
 and a great part of it isin verse. In the ¢riads, for instance : 
 
 Efficiunt totum Casus, Natura, Voluntas, 
 
 Dat triplicem mundum Deitas, Natura, Mathesis, 
 
 Hine tria principia emanant, Lux, Spiritus, Unda, 
 
 Est animus triplex Vita, Sensu, Ratione. 
 M. Librihas quoted at length (vol. iv. note x.) the Copernican chapter 
 of the work de Immenso. 
 
 Frankfort, fifteen-nimety-two. Peter Ramus and Laza- 
 rus Schoner. ‘Petri Rami Arithmetices Libri Duo, et Al- 
 gebree totidem: a Lazaro Schonero emendati et explicati. 
 Ejusdem Schoneri libri duo: alter, De Numeris figuratis ;, 
 alter, De Logistiea Sexagenaria,’ Octavo (Q). 
 
 The first edition of Ramus is said to be of fifteen-eighty-four. 
 P.'427. 
 
 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 (Q). 
 
 The first book and the addition are on abstract numbers and 
 fractions: the second book is on commercial arithmetic: the third 
 book is on the extraction of roots, on surds, and on cossic num-. 
 bers, or algebra. Masterson must have been a valuable help to the 
 student: though he would have been more so if he had used the 
 
 modern method of division. 
 ; DZ 
 
30 A CHRONOLOGICAL LIST OF 
 
 London, fifteen-ninety-four. Blundevile. ‘M. Blunde- 
 vile His Exercises, containing sixe Treatises,’ &c. Quarto in 
 fours. 
 
 He was the first introducer of a complete trigonometrical canon 
 into English, the first announcer of Wright’s discovery of meri- 
 dional parts, and he exercised much influence on the studies of 
 the first part of the seventeenth century. Nor is he altogether 
 out of date yet; for I lately observed (Mechanic’s Magazine, vol. 
 xliv. p. 477) that a patent for an improvement in horse-shoes was 
 upset in the Court of Chancery, on proof that Blundevile had de- 
 scribed it in one of his books on horses, to which he refers 
 several times in the present work. ‘The first of these exercises is 4 
 treatise on arithmetic, in the form of question and answer, written 
 for Elizabeth Bacon, the sister of the great philosopher. It is a 
 dogmatical treatise, sufficiently clear. The seventh edition of these 
 exercises was published in sixteen-thirty-six, and I believe there 
 was none later. 
 
 London, fifteen-ninety-six. ‘The Pathway to Knowledge. 
 Conteyning certaine briefe Tables of English waights, and Mea- 
 sures .... 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 (0). 
 
 This work has escaped the notice of Ames, and was perhaps 
 confounded with the geometrical work of the same name by Recorde. 
 The English translator’s preface, giving an account of the existing 
 weights and measures, is the most complete thing there is of the 
 day. The arithmetic is not equal to that of Recorde or Masterson, 
 nor the book-keeping to that of Mellis, aud the algebra is far be- 
 hind Recorde’s. The translator gives the following verses, the first 
 of which are now well known. I suspect he is the author of them, 
 having never seen them at an earlier date: Mr. Halliwell, who is 
 more likely than myself to have found them if they existed very 
 early, names no version of them earlier than 1635 : 
 
 Thirtie daies hath September, Aprill, June, and November. 
 Febuarie eight and twentie alone, all the rest thirtie and one. 
 
 Looke how many pence each day thou shalt gaine, 
 Just so many pounds, halfe pounds and groates : 
 with as many pence in a yeare certaine, 
 
 Thou gettest and takest, as each wise man notes. 
 
 Looke how many farthings in the weeke doe amount. 
 In the yeare like shillings, and pence thot shalt count. 
 
 To complete this subject: Mr. Davies [Key to Hutton’s Course, 
 
WORKS ON ARITHMETIC, 31 
 
 p- 17] quotes the following from a manuscript of the date 1570, or 
 near it: 
 
 Multiplication is mie vexation, 
 
 And Division is quite as bad, 
 
 The Golden Rule is mie stumbling stule, 
 
 And Practice drives me mad. 
 
 Alcmaar (preface dated from Amsterdam), fifteen-ninety- 
 six. Nicolaus Petri. ‘Practicque omte Leeren Rekenen 
 Cypheren ende Boeckhouwen met die Reghel Coss... .’ 
 Octavo (small) (O). 
 
 Contains arithmetic, algebra, geometry, and an example of book- 
 keeping. 
 
 Venice, fifteen-ninety-nine. Joh. Bapt. Benedictus. 
 ‘Speculationum Liber.’ Folio in twos. 
 
 The first speculation is entitled ‘ Theoremata Arithmetica,’ and 
 is a laboured explanation of the principles of arithmetical rules by 
 reference to geometry, beginning with the old difficulty of the pro- 
 duct of two fractions being less than either of the factors. 
 
 London, sixteen-hundred. ‘Thomas Hylles. ‘The arte 
 of vulgar arithmeticke, both in integers and fractions, devided 
 into two bookes . . . . Nomodidactus Numerorum .. . . Por- 
 tus Proportionum .... whereunto is added a third book en- 
 tituled 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 (QO). 
 
 This is in dialogue ; but whenever any rule or theorem is deli- 
 vered, it is in verse. It is a big book, heavy with mercantile lore. 
 The following are specimens of the verses : 
 
 Number is first divided as you see 
 For number abstract, and number contract 
 And numbers abstract are such as stand free 
 From every substance so cleare and exact 
 That they have no sirname or demonstration 
 Save the pure units of their numeration, : 
 
 All primes together have no common measure 
 Exceeding an ace which is all their treasure. 
 
 Addition of fractions and likewise subtraction 
 Requireth that first they all have like basses 
 Which by reduction is brought to perfection 
 And being once done as ought in like cases, 
 Then adde or subtract their tops and no more 
 Subscribing the basse made common before. 
 
 The partition of a shilling into his aliquot parts. 
 
 A farthing first findes fortie eight 
 An halfepeny hopes for twentie foure 
 
32 A CHRONOLOGICAL LIST OF 
 
 Three farthings seekes out 16 streight 
 A peny puls a dozen lower. 
 
 Dicke dandiprat drewe 8 out deade 
 Twopence tooke 6 and went his way 
 Tom trip and goe with 4 is fled 
 
 But goodman grote on 3 doth stay 
 
 A testerne only 2 doth take 
 
 Moe parts a shilling cannot make. 
 
 In spite of all this trifling, Hylles was a man of learning. He 
 cites both Lucas de Burgo and Peter de Burgo. 
 
 Paris, sixteen-hundred. John Chambers. ‘Barlaami 
 Monachi Logistica nune primum Latiné reddita et scholiis 
 illustrata.? Quarto. 
 
 The Greek is given at the end. Barlaam lived in the fourteenth 
 century, and his work is mostly on fractions and on proportions. 
 See the Penny Cyclopedia, ‘ Barlaam,’ for information on the author 
 and on this edition. 
 
 Leyden, sixteen-three. ©. Dibuadius. ‘In Geome- 
 triam Euclidis prioribus sex Elementorum libris comprehensam 
 Demonstratio Numeralis.’ Quarto. 
 
 The six books of Euclid are, for the most part, here verified by 
 arithmetical or trigonometrical instances. The usual mode, or linear 
 demonstration, is made to follow in another book. 
 
 Leyden, sixteen-five. Simon Stevinus. ‘ Tomus secun- 
 dus Mathematicorum hypomnematum de Geometrize praxi.’ 
 Folio in threes. (The printing delayed beyond the title-date.) 
 
 This work contains the celebrated Latin treatise on book-keeping 
 (Peacock, Encyl. Metrop. ‘ Arithmetic,’ p. 464), in which the diffi- 
 culty of finding Latin renderings of the technical terms is well got 
 over. All writers attribute this translation to the celebrated Wille- 
 brord Snell, who passed from fourteen to eighteen years of age while 
 the work was printing. Now the fact is, that not only does Stevinus 
 date his own preface to this treatise on book-keeping in August 1607, 
 but at the end of the volume he excuses himself from giving several 
 matters which had been announced at the beginning, because the 
 printer was tired of waiting, and he had not satisfied himself about 
 them. And in this same final notice he mentions Will. Sn. (Snell) 
 as a person who had written a letter for him : styling him mathema- 
 tum neque ignarus neque expers ; much too poor a compliment for a 
 youth who, at such an age, had translated, and therefore understood, 
 all the writings of Stevinus. The story about the translation is from 
 Gerard Vossius, who was afterwards Snell’s colleague, and who 
 made a mistake and confusion of persons which was copied by Bayle, 
 Beckmann, &c. M. Quetelet has recently published a notice of 
 Stevinus, which states that he was born in 1548, and died in 1620. 
 P2464, 
 
WORKS ON ARITHMETIC, 33 
 
 Augsburgh, sixteen-nine. Geo. Henischius. ‘ Arith- 
 metica perfecta et demonstrata.’ Quarto (ON). 
 
 A laboured work in seven books. Its algebra is that of a former 
 day ; its power of computation very fair. Dr. Peacock refers se- 
 veral times to another work of Henischius, ‘ De numeratione multi- 
 plici,’ sixteen-five. 
 
 Mayence, sixteen-eleven. Christopher Clavius. ‘ Epi- 
 tome Arithmetices Practicee.’ Folio in threes (O). 
 
 This is in the second volume of Clavius’s collected works. [It 
 is said to have been first published in fifteen-eighty-three.] Perhaps 
 there are more extensive examples of the square root worked by the 
 
 old method in this treatise than would easily be found elsewhere. 
 P. 426. 
 
 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 a+ rs: &c. He 
 
 then uses these fractions in approximation, but without the assist- 
 ance 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 
 3073763825935885490683681 __ 
 3695489834122985502126240 
 
 London, sixteen-thirteen. John Tap. ‘The Pathway to 
 Knowledge,’ &c. Octavo (QO). 
 
 This professes to be a reprint of the work published under that 
 name (1596), and above described. But it is in fact the substance 
 of the above work thrown into the form of dialogue, with the book- 
 keeping reprinted, and some algebra taken from V. Menher above 
 mentioned. 
 
 London, sixteen-thirteen. Richard Witt. ‘ Arithmeti- 
 call 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) (QO). 
 
34 A CHRONOLOGICAL LIST OF 
 
 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 frac- 
 tions are really used: the tables being constructed for ten mil- 
 lion of pounds, seven figures have to be cut off, and the reduction 
 to shillings and pence with a temporary decimal separation, is intro- 
 duced when wanted. For instance, when the quarterly table of 
 amounts of interest at ten per cent is used for three years, the prin- 
 cipal being 100/. (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 
 
 1 1372 | 66429 
 Facit < sh 13 | 2858. 
 d 31 4296.? 
 
 Giving 1372]. 13s. 3d. for the answer, And the tables are 
 expressly stated to consist of numerators, with 100... for a deno- 
 minator. 
 
 [A reprint of this work, by T. Fisher, Zondon, sixteen-thirty- 
 four, duodecimo, is in the Royal Society’s Library. | 
 
 Paris, sixteen-fourteen. Artabasda. ‘Nic. Smyrnei 
 Artabasdze Greeci Mathematici EK®PASIC Numerorum Nota- 
 tionis per gestum digitorum. Item Venerab. Bedze de Indi- 
 gitatione et manuali loquele lib.’ Octavo in twos (edited by 
 F. Morell, the first Gr. Lat., the second Lat.). 
 
 These tracts are on nothing but the mode of representing nu- 
 merals by the fingers. Bede’s tract is not a separate work, but a 
 chapter from his treatise de natura rerum, in which it will be found 
 in the ‘ Bede . . . opuscula plura,’ edited by Noviomagus, Cologne, 
 fifteen-thirty-seven, folio in twos. But the first edition is, Venice, 
 fifteen-twenty-five, Joh. Tacuinus (editor and printer), ‘hoc in vo- 
 lumine hee continentur M. Val. Probus..... Ven. Beda de Com- 
 puto per gestum digitorum ....nune primum edita.’ Quarto in 
 Sours. 
 
 Paris, sixteen-fourteen. Eionorat Meynier. ‘L’ Arith- 
 metique.... enrichie de ce que les plus doctes mathemati- 
 ciens 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 (OQ). 
 
 There is a great deal of matter in this book, which runs to 664 
 pages; and it might be historically useful. Part of it is an attack 
 
WORKS ON ARITHMETIC. 35 
 
 on Stevinus, who, by the way, is represented as then alive. The 
 following is one of the examples in subtraction: ‘ L’an 1535, Jean 
 Calvin composa son labirinthe abominable (puisque dans iceluy s’est 
 perdu un grand nombre d’hommes de bien) . . . je demande com- 
 bien s’est passé d’années depuis qu’il composa le dit Dedale.’ 
 
 London, sixteen-fourteen. ‘Thomas Bedwell. ‘De Nu- 
 meris Geometricis. Of the nature and properties of geome- 
 trical numbers, first written by Lazarus Schonerus, and now 
 Englished,’ &e. Quarto. 
 
 On figurate, square, &c. numbers, with applications to men- 
 suration. 
 
 Bologna, sixteen-sixteen. P. A. Cataldi. ‘ Quarta 
 parte della pratica Aritmetica Dove si tratta della principalis- 
 sima, et necessarlissima legola chiamata comunemente del 
 Tre,’ &c. Folio (quarto siz2). 
 
 A long and wordy treatise on the rule of three, but venturing on 
 examples of a much higher order of difficulty of computation than 
 had been previously attempted. 
 
 Edinburgh, sixteen-seventeen. JohnWNapier. ‘ Rab- 
 dologize seu Numerationis per Virgulas Libri duo: cum Ap- 
 pendice de expeditissimo Multiplicationis Promptuario, Quibus 
 accessit et Arithmetice Localis Liber unus. Authore et in- 
 ventore Johanne Nepero,’ &c. Duodecimo (QO). 
 
 A posthumous work. Napier’s rods are well known, and this 
 book is the descriptio princeps of them, with applications. The use 
 of decimal fractions, expressly attributed to Stevinus, renders it re- 
 markable. It is stated (P. 441) that Napier invented the decimal 
 point, but this is not correct : 1993-278 is written by him 19932/73”. 
 
 Oppenheym, sixteen-seventeen and nineteen. Robert 
 Fludd. ‘ Utriusque Cosmi Majoris scilicet et minoris meta- 
 
 physica, physica atque technica historia.” Two volumes. Folio 
 in twos. 
 
 The second volume has another title-page, purporting that it 
 gives the supernatural, natural, preternatural, and contranatural his- 
 tory. Of Robert Fludd (though he was great enough in his day to 
 engage the attention of Descartes) I shall here say nothing, except 
 that he gave his mixture of mysticism and science two dedications, 
 one on each side ofaleaf. The first, signed Ego, homo, was addressed 
 to his Creator: the second, signed ‘ Robert Fludd’ to James I. of 
 England. The first volume contains a treatise on Arithmetic, and 
 on Cossic Arithmetic, or Algebra. The arithmetic is rich in the 
 description of numbers, the Boethian divisions of ratios, the musical 
 system, and all that has any connexion with the numerical mysteries 
 
36 A CHRONOLOGICAL LIST OF 
 
 of the sixteenth century. The algebra is of the four rules only, 
 referring for equations and other things to Stifel and Recorde. The 
 signs of addition and subtraction are P and M with strokes drawn 
 through them. The notation for powers is the q, c, qq, &c. series of 
 symbols. Perhaps the most remarkable thing about the algebra is, 
 that Fludd, who wrote it in France for the instruction of a Duke of 
 Guise, should have known nothing of Vieta. The second volume is 
 strong upon the hidden theological force of numbers. 
 
 London, sixteen-nineteen. Hfenry 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 (OQ). 
 
 Brasser seems to have been a celebrated teacher: it is not clear 
 whether he published or not. The book itself is a mixture of arith- 
 metic and algebra in Scheubel’s form, with much of algebraical 
 application to commercial questions. There is extraction of roots as 
 far as the fourth. 
 
 Hamburg, sixteen-twenty-one. Peter Lauremberg of 
 Rostoch. ‘ Institutiones Arithmetice.’ Octavo (small). 
 
 A demonstrative book. 
 
 Rostoch, sixteen-twenty-one. John Iauremberg. ‘Or- 
 ganum analogicum, sive instrumentum proportionum.’ Quarto. 
 
 This instrument is the sector. 
 
 Rostoch, sixteen-twenty-three. John Lauremberg. 
 ‘ Virgularum numeratricum et promptuarii arithmetici deserip- 
 tio, figuree 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). 
 
 There is no hint here of decimal fractions. Attached is a ‘ choro- 
 logicall discourse’ on the management of estates. 
 
WORKS ON ARITHMETIC. 37 
 
 London, sixteen-twenty-eight. John Speidell. ‘An 
 Arithmeticall Extraction or Collection of divers questions with 
 their answers.’ Octavo (small). 
 
 For John Speidell, see the article Tables in the Supplement of 
 the Penny Cyclopedia. The book is literally what it professes to 
 be, questions with their facits or answers, and nothing else, mostly 
 in reduction, the rule of three, and practice. 
 
 Amsterdam, sixteen-twenty-nine. Albert Girard. ‘In- 
 vention nouvelle en l’algebre,’ &c. Quarto. 
 
 In this celebrated work there is a slight treatise on arithmetic, 
 the most remarkable part of which is, that the author gives no ex- 
 amples in division by more than one figure, and seems to decline 
 them as too difficult for his readers: he gives some results only. 
 P. 426, 442. On one occasion he uses the decimal point. 
 
 Oughtred’s Clais is a book of arithmetic as well as of al- 
 gebra, and one of great:celebrity. The editions that have fallen in 
 my way are: 
 
 London, sixteen-thirty-one. ‘ Arithmeticee in numeris et 
 speciebus imstitutio: quee tum logisticze, tum analyticee, atque 
 adeo totius mathematicze, quasi clavis est.’ Signed ‘ Guilel- 
 mus Oughtred,’ at the end of the dedication. Octavo. 
 
 London, sixteen-forty-seven. ‘The Key of the Mathema- 
 ticks New Forged and Filed: Together with A Treatise of the 
 Resolution of all kinde of Affected Aiquations in Numbers. 
 With the Rule of Compound Usury; And demonstration of 
 the Rule of false Position. And a most easie Art of delineat- 
 ing all manner of Plaine Sun-Dyalls. Geometrically taught 
 by Wil. Oughtred.’ Octavo. 
 
 Oxford, sixteen-fifty-two. ‘ Guilelmi Oughtred...... 
 Clavis Mathematicee denuo limata, sive potius fabricata... . 
 Editio tertia auctior et emendatior. Octavo. 
 
 Oxford, sixteen-sixty-seven. ‘Do. do..... Editio quarta 
 auctior et emendatior.’ Octavo. 
 Oxford, sixteen-ninety-three. ‘Do. do..... Editio quinta 
 
 auctior et emendatior.’ Octavo. 
 
 Ozford, sixteen-ninety-eight. ‘Do. do. Editio quinta 
 auctior et emendatior. Ex recognitione D. Johannis Wallis, 
 S.T.D. Geometriz Professoris Saviliani.’ Octavo. 
 
 London, sixteen-ninety-four (again in seventeen-two). ‘Mr. 
 William Oughtred’s Key of the Mathematics. Newly trans- 
 lated . ..:with Notes ...... Recommended by Mr. E. Hal- 
 ley.’ Octavo. 
 
 The third edition was an extended re-translation into Latin; and 
 
 E 
 
38 A CHRONOLOGICAL LIST OF 
 
 the preface of it, which was copied into succeeding editions, shews 
 that it was carefully performed under Oughtred’s own eye. The 
 treatise on dialling was translated into Latin by a young man of 
 sixteen, of Wadham College, whom Oughtred describes as already 
 an inventor in Astronomy, Gnomonics, Statics, and Mechanics, and 
 of whose future fame he augurs great things: his name was Chris- 
 topher Wren. ‘There are two fifth editions in the list, the second of 
 them revised, it appears, by Dr. Wallis, who ought, one would think, 
 to have known better the history of the book he edited. But on 
 examining them, I find that they are the same impressions with 
 different title-pages : so that it would seem as if Wallis had allowed 
 his eminent name to appear as a guarantee for a book he had never 
 revised. Wrong again, however: for in the preface to the third 
 edition it appears that Wallis had given all the help of an editor 
 at that stage of the work; so that his name should rather have 
 appeared before. 
 
 The editions which follow the first, besides other additions, have 
 the solution of adfected Equations, or Vieta’s method (see the Index, 
 ‘Horner’), Throughout, the old, or scratch method of division, 
 is retained. Dr. Peacock observes of this method that it lasted 
 nearly to the end of the seventeenth century: but it thus appears 
 that it even got into the eighteenth, and that with Halley’s name as 
 arecommendation. Throughout all the editions, Oughtred’s ancient 
 algebraical notation is retained, as also his way of writing decimal 
 fractions (12/3456 for our 12°3456). I cannot tell why Dr. Peacock 
 
 always spells his name Oughtrede (p. 441, &c.). 
 
 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. 
 
 The second edition, Oxford, sixteen-sixty, ociavo, is ‘ by the 
 Author’s consent Revised, Corrected .... by A. H. Gent.’ [The 
 editor was Arthur Haughton.] The addition above named gives 
 the first description of the sliding rule. The circles of proportion 
 are what would now be called circular sliding rules; and the two 
 rulers are the common sliding rule, the rulers being kept together 
 by the hand. Various errors are afloat about the invention of the 
 sliding rule. Some give it to Wingate, some to Gunter, some to 
 Partridge; the official book of the Excise Office gives it to an excise 
 officer whose name I forget. But the truth is that Oughtred in- 
 vented it. (See Penny Cycl. ‘Slide Rule.’) 
 
 London, sixteen-thirty-three. Robert Butler. ‘The 
 
WORKS ON ARITHMETIC. 39 
 
 Scale of Interest ; or Proportionall Tables and Breviats .... . 
 Together with the valuation of Annuities,’ &e. Octavo. 
 
 These tables are for discount or present value; and as far as 
 annuities, &c. are concerned, that is, in compound-interest questions, 
 resemble Witt’s. But decimals are really used in the applications, 
 the separating notation being as in 12|3456. 
 
 London, sixteen-thirty-three. Wicholas Hunt, M.A. 
 ‘The Hand-Maid to Arithmetick refined: Shewing the variety 
 and facility of working all Rules in whole Numbers and Frac- 
 tions, 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 use- 
 full for all Gentlemen, Captaines, Gunners, Shopkeepers, Arti- 
 ficers, and Negotiators of all sorts: Rules for Commutation 
 and Exchanges for Merchants and their Factors. A Table 
 from 1]. to 100 thousand, for proportionall expences, and to 
 reserve for Purchases.’ Octavo (ON). P. 442. 
 
 The first thing that strikes any one in this book is the slavish- 
 ness of the dedication to an Earl, and the grotesque appearance 
 which the use of the ambiguous word rare gives it. ‘ When I re- 
 flect. on ancient Nobility, earth’s glory, it being found in the way of 
 vertue ; this so rare a thing transports my soule with thoughts of a 
 glorious eternitie. And for as much as the Lord hath said (Dixit 
 dii estis) that you are titular and tutelar terrestriall gods, powerfully 
 protecting the meaner shrubbes under the spreading branches of 
 your tall Cedars: I wonder not to see almost every Pamphlet to 
 crouch and deject it selfe in all humility and lowlinesse, seeking 
 patronage from persons of eminency.” He afterwards calls on his 
 patron to ‘imitate the propitiousnesse of the divine essence,’”’ which 
 will excite him to ‘a voluntarie prostitution of his humble service,’’ 
 if the patron will protect him from ‘‘ that squint-ey’d foole, that 
 antipathie to vertue (the envious man).” 
 
 The book itself is very full on weights and measures, and com- 
 mercial matters generally. It does not treat of decimal fractions : 
 what the author calls ‘decimall Arithmeticke’ is the division of a 
 pound into 10 primes of two shillings each; each shilling into six 
 primes of two pence each. There are some verses, perhaps sug- 
 gested by the republication of Buckley, headed: ‘ Arithmeticke- 
 Rithmeticall, or the Handmaid’s Song of Numbers;’ of which the 
 rules of addition and subtraction will give a sufficient specimen : 
 
 Adde thou upright, reserving every tenne, 
 And write the Digits downe all with thy pen, 
 The proofe (for truth I say,) 
 Is to cast nine away. 
 From the particular summes, and severall 
 Reject the Nines; likewise from the totall 
 When figures like in both chance to remaine 
 Clear light of working right shal be your gain; 
 
40 A CHRONOLOGICAL LIST OF 
 
 Subtract. the lesser from the great, noting the rest, 
 Or ten to borrow, you are ever prest, 
 
 To pay what borrowed was thinke it no paine, 
 But honesty redounding to your gaine. 
 
 Paris, sixteen-thirty-four (with another title-page, sixteen- 
 forty-four, bat it is all one impression). Peter Herigone. 
 ‘Cursus Mathematici tomus seeundus.” Octavo (QO). 
 
 The volumes of this course, which is said to be the first com- 
 plete 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 dela 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 : 
 
 (‘4 B C D de F G L M N 
 
 1 2 3 4 5 6 t 8 9 0 
 
 a e i oO ar er ir or ur 
 ra re rh ro ru ¢ 
 
 So that r signifies five except before or after wu, when it destroys the 
 meaning altogether. Grey’s system is: 
 
 B D TF F L S P K N b 
 Z 8 9 
 a e i o u au Oi ei ou 
 
 The vowel-systems of the two might be combined. 
 
 London, sixteen-thirty-four. William Webster. ‘ Web- 
 ster’s tables for simple interest direct . . . . Also his Tables for 
 Compound Interest...” Octavo (small). Third edition. 
 
 This work treats decimal arithmetic as a thing known: but all 
 the tables are not decimals. Neither is there yet any recognised 
 decimal pomt; only a partition-line to be used on occasion. In this 
 book I find the first head-rule for turning a decimal fraction of a 
 pound into shillings, pence, and farthings: though not so perfect a 
 one as was afterwards found. I have seen the second edition, place, 
 title, and form as above, sixteen-twenty-nine. The copy of each 
 edition which I have examined is bound up with King James’s list 
 of customs, ‘The Rates of Marchandizes’ &c. promulgated in 1605. 
 
 London, sixteen-thirty-seven (again in forty-seven). Leo-= 
 nard Digges. ‘A book named Tectonicon .... Published 
 by Leonard Digges, Gentleman, in the yeere of our Lord 
 15562 Quarto. 
 
WORKS ON ARITHMETIC. 41 
 
 A book of mensuration : one of the instances in which a book was 
 republished for the name of its author, long after its methods were 
 obsolete. This occurred so frequently in the seventeenth century 
 that we must take for granted the existence of a class of mathema- 
 ticians to whom the light of Napier, &c. had not penetrated. 
 
 Loxdon, 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 Avoir- 
 dupois Weights. Containing divers Orders and Articles made 
 and set forth by the Right Hon”* the Lords and Others of his 
 Majesty’s most Hon’® Privy Council ...... Published by 
 their Lordships Orders.’ Quarto. 
 
 This does not look like a book of arithmetic; but Penkethman 
 has thought it necessary to prefix instruction in numeration, both 
 Roman and Arabic. 
 
 ‘Note that 1v signifies 1111, as 1x signifies nine which takes 
 as it were by stealth, or pulls back one from foure and ten.’ So 
 that, in fact, 1 stands behind x and picks his pocket. 
 
 [ 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. 
 
 Herborne 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. ‘ Anth- 
 metica. . . . exemplis historicis illustrata, itidem algebree prin- 
 cipia.” 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. ‘ Tay xara 
 MOSNLATINGY YONT WAV €i¢ chy rov WAarwvos avayreow. Quarto. 
 Edited by Ismael Bullialdus (Bouillaud). 
 
 Leyden, eighteen-twenty-seven. Theon of Smyrna. The 
 same title. Octavo in iwos. Edited by J. J. de Gelder. 
 
 These are the only editions of Theon of Smyrna, neither having 
 the complete work, if indeed any complete manuscript exist. The 
 first has the arithmetic and music; the second, with various read- 
 ings, the arithmetic only. The arithmetic is nothing more than 
 
 E2 
 
49 A CHRONOLOGICAL LIST OF 
 
 that classification and nomenclature of numbers which is also given 
 by Nicomachus (the two were contemporaries), and followed by 
 Boethius. The music, as in other cases, is only discussion of frac- 
 tions under names derived from: the scale. Theon of Smyrna must 
 not be confounded with his greater namesake of Alexandria. 
 
 London, sixteen-forty-five. Edmund Wingate. ‘Use 
 of the Rule of Proportion in Arithmetique and Geometrie.’ 
 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 Ma- 
 thematica.’ Folio in twos. 
 
 In this collection is the ‘De numerosa potestatum purarum at- 
 que adfectarum ad exegesin resolutione tractatus,’ which was first 
 published, Paris, sixteen-hundred, folio, and is the first exposi- 
 tion of the method by which the general evolution of the roots of 
 equations is connected in operation and principle with the extraction . 
 of the square root and common division. See the references pre- 
 sently given under the name of Horner. The first departure from 
 Vieta’s form of solution, which amounts to incorporating with it the 
 method afterwards known by the name of Newton, was made by 
 Henry Briggs in the ‘Trigonomet:ia Britannica,’ Gouda, sixteen- 
 thirty-three, folio in twos and threes. Accordingly, Briggs has a fair 
 claim to be the first user of that which is called Newton’s method of 
 approximation, though he does not explain it, but leaves the ex- 
 planation to be collected from his examples. The work which in- 
 troduced Vieta’s method into England was the ‘ Artis Analytice 
 Praxis’ of Thomas Harriot, London, sixteen-thirty-one, folio. All 
 these writers would have called their process algebraical, but the 
 progress of the art of computation will make their works essential 
 parts of the history of arithmetic. I believe the most complete 
 account of Vieta is that which I have given in the Penny Cyelo- 
 pedia : with which may be read the account of his works in Hutton’s 
 Tracts, vol. ii. pp. 260-274. 
 
 London, sixteen-forty-eight. Seth Partridge. <‘ Rabdo- 
 logia, or the Art of numbring by Rods.’ Duodecimo (0). 
 
 The use of Napier’s rods or bones explained. 
 
 Napier’s Rabdologia brought the well-known Napier’s rods into 
 vogue for half a century (P. 411). And the reason seems to have 
 
 been that this contrivance really was found useful, in the state of the 
 habit of computation as it then existed. P. 432, 440. 
 
 Leyden, sixteen-forty-nine. Joh. Hen. Alsted. ‘Scien- 
 tiarum omnium Encyclopedia.’ Two volumes. Folio in threes. 
 
 The first edition is said to be sixteen-thirty, and the preface of 
 
WORKS ON ARITHMETIC. 43 
 
 this one is dated sixteen-twenty-nine, but it refers to a course of phi- 
 losophy published in sixteen-twenty, of which it seems to be an 
 enlargement. As far as I can learn, this is the first of the works to 
 which moderns attach the idea of a Cyclopedia. For though Ringel- 
 berg and Martinius had published under that name, and though the 
 Margarita Philosophica above mentioned and other works ran through 
 the whole circle of elementary knowledge—yet I cannot establish 
 that any work before Alsted’s was so large and comprehensive as to 
 be a Cyclopedia in the sense in which Chamber’s Dictionary was 
 afterwards so called. The course of arithmetic (O) in the second 
 volume is rather scanty, but seems less so when the bareness of the 
 professed mathematical courses is considered. There isa table of 
 the squares and cubes of all numbers up to 1000. Logarithms are 
 not mentioned, but Napier’s rods are. Moreover, by the date of 
 the first edition, we must call this a course of mathematics prior to 
 Herigone’s, usually called the first. 
 
 London, sixteen-fifty. John Wyhbard.  ‘ Tactometria. 
 seu, Tetagmenometria. Or, The Geometry of Regulars practi- 
 cally 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 Arith- 
 metick: Discovering the secrets of that Art, in Numbers and 
 Species.’ Octavo (N). 
 
 This is Jonas Moore’s first work, and is a very good one. It is 
 very complete in decimals, giving the contracted multiplication and 
 division. It has the use of logarithms (in which the operations with 
 the negative characteristic are fully given), algebra from Oughtred, 
 and squares and cubes of all numbers up to 1000, and the fourth, 
 fifth, and sixth powers up to 200. The second edition is: 
 
 London, sixteen-sixty. ‘ Moor’s Arithmetick in two books.’ 
 Octavo. 
 
 It is worthy of remark that the common arithmetic of this edition 
 is in black letter, the second book, or Algebra, being Roman letter, 
 as all the previous edition was. Perhaps Recorde, &c. had given 
 common readers a prejudice for black letter arithmetics. P. 442. 
 
 Dantzig, sixteen-fifty-two. Joh. Broscius. ‘ Apologia 
 pro Aristotele et Euclide ... . . Additze sunt duze discepta- 
 tiones de numeris perfectis.’ Quarto. 
 
 A perfect number is one which is equal to the sum of its less fac- 
 tors: thus 28, being the sum of 1, 2, 4, 7, and 14, is a perfect num- 
 ber. And a number was called defective or redundant, according 
 as the sum of its factors fell short of or exceeded the number. Thus 
 8 is defective, and 12 is redundant. 
 
44 A CHRONOLOGICAL LIST OF 
 
 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 Mer- 
 catoria, Arithmetick Natural and Decimal,’ &c. Octavo (N). 
 
 Dr. Peacock (p. 452) has given the first book notoriety by his 
 citations from the eulogistic poetry at the beginning, which he has 
 by no means exhausted. The poem of Geo. Wharton, the royalist 
 astrologer, on the existing state of things, is a political satire. An- 
 other commends the plainness of the work, hints at decimals and 
 logarithms becoming too common, and pronounces that a merchant 
 
 ‘may fetch home the Indies, and not know 
 What Napier could, or what Oughtred can doe.’ 
 
 Dr. Peacock speaks lightly of the work, in mentioning the excessive 
 praises with which it was announced. But a very short treatise, 
 explicit upon the modern mode of division, which those who praised 
 it had perhaps never seen before, and upon the use of practice, was 
 rather an exception to the rule at the time at which this book was 
 published. 
 
 The second work is more learned, and has an appendix on deci- 
 mals. The author disapproves of the use which some would make 
 of decimals, and avers that the rule of practice is more convenient 
 in many cases; which is perfectly true. Bridges seems to have had 
 a very clear view of the capabilities of arithmetic without formal 
 fractions. He mentions some preceding authors, and his list is ‘The 
 Merchant’s Jewel,’ 1628; ‘The Handmaid to Arithmetick,’ 1638 ; 
 ‘The Map of Commerce,’ 1638; Masterman, Johnson, Hill, Ley- 
 bourn, Recorde, Baker, Val. Menher, Boteler. 
 
 London, [sixteen-fifty-four]. E[dmund] Wingate]. 
 ‘Ludus Mathematicus. Or, The mathematical Game: Explain- 
 ing the description, construction,, and use of the Numericall 
 Table of Proportion.’ Duodecimo. (Again in eighty-one.) 
 
 This is the description ofa logarithmic instrument, which it would 
 be impossible to give a notion of without the instrument itself, or a 
 drawing. ‘The date is cut out in my copy. 
 
 London, sixteen-fifty-five. ‘Thos. Gibson. ‘ Syntaxis 
 Mathematica.’ Octavo. 
 
 This work is filled with the construction of equations from Des 
 Cartes; but it gives the arithmetical solution, or Vieta’s Exegesis, 
 from Harriot : also interest-tables, partly from Simon Stevens. 
 
 Oxonii, sixteen-fifty-seven. John Wallis. ‘ Mathesis 
 Universalis: sive Arithmeticum Opus Integrum,’ &c. Quarto. 
 
 This and other works, with their separate title-pages, are col- 
 lected (apparently so published) as Opera Mathematica, and were 
 
WORKS ON ARITHMETIC. 45 
 
 afterwards republished, in substance, in the folio collection of Wal- 
 lis’s works. The Arithmetic gives both modes of division: but the 
 old notation, as 12/345, is used for decimal fractions. Wallis after- 
 wards adopted the decimal point in his Algebra. 
 
 Wurtzburg, sixteen-fifty-seven-and-eight. Gaspar Schott. 
 ‘ Magia Universalis Nature et Artis.’ Quarto (four parts, va- 
 riously bound in volumes). 
 
 This work, and others of a similar character by the same author, 
 are the precursors of the works of mathematical amusement com- 
 piled by Ozanam, Montucla, Hutton, &c. In the fourth part he 
 treats of arithmetic, mostly on the wonders of combinations. The 
 gem of the book is his accurate calculation of the degrees of grace 
 and glory of the Virgin Mary, which are exactly 
 
 115792089237316195423570985008687 9078532 
 69984665640564089457584007913129639936 
 
 not one more nor less. This is the 256th power of 2, and is repeated 
 three times, printed in the same way. Others solved the same 
 problem, and found out the number of stars at the same time, by 
 writing down every possible way in which the words of 
 
 Tot tibi sunt dotes, Virgo, quot sidera ccelo 
 
 can be arranged in an hexameter line. 
 
 As Schott here describes the digit of his geometrical pace as 
 being forty poppy-seeds placed side by side, I thought it possible I 
 might get a fellow to the barley measure (see page 8). But it 
 seems that he has not played fair, or has thought it necessary to 
 bring the degrees of glory of the Virgin up to the 256th power of 2, 
 at any sacrifice. For I find that forty poppy-seeds, side by side, 
 measure an inch and a half (English); so that the geometrical foot 
 of sixteen digits would be two feet English. 
 
 Leyden, sixteen-sixty. Wincent Leotaud. ‘ Institu- 
 tionum arithmeticarum libri quatuor, in quibus, omnia quee ad 
 numeros simplices, fractos, radicales, et proportionales perti- 
 nent precepta clarissimis demonstrationibus tum arithmeticis 
 tum geometricis illustrata traduntur.’ Quarto (QO). 
 
 This is a clear but heavy work, running to 700 pages. 
 
 Herbipoli(Wurtzburg), sixteen-sixty-one. Gaspar Schott. 
 ‘Cursus Mathematicus.’ Folio in threes. 
 
 It has a shabby arithmetic, which, considering the magnitude of 
 the course, reminds us of Falstaff’s halfpennyworth of bread. In 
 the frontispiece, a lion and a bear draw a car surmounted by an 
 armillary sphere, and having for its wheels (or rather castors) the 
 celestial and terrestrial globes, along an amphitheatre the pavement 
 of which is studded with diagrams. These diagrams contain a sly 
 
46 A CHRONOLOGICAL LIST OF 
 
 little bit of freemasonry. Schott, as a good Jesuit, adheres to the 
 Ptolemaic system, grudgingly, and because his church had condemned 
 the other, as appears from his words. But in the diagrams on the 
 pavement of his frontispiece, he draws the Copernican system, and 
 no other. Dr. Peacock (p. 408) refers to the ‘ Arithmetica Prac- 
 tica’ of Schott [ Herb. sixteen-sixty-two—— ]. 
 
 London, sixteen-sixty-four. James Hodder. ‘Hodder’s 
 Arithmetick: or, That necessary Art made most easie.’ The 
 third edition, much enlarged. Duodecimo (0). 
 
 Had this work given the new mode of division, it must have 
 stood in the place of Cocker. The ninth edition was published in 
 sixteen-seventy-two. [The first edition is said to have had the date 
 1661.] In Davis’s sale catalogue (1686), there are marked as by 
 Hodder, a Vulgar Arithmetic, 1681, and a Decimal Arithmetic, 
 1671. There are no decimal fractions in the book before us. 
 
 Deventer, sixteen-sixty-seven. Joachim Camerarius. 
 ‘Explicatio in duos libros Nicomachi Geraseni.. . . et note 
 Samuelis Tennulii in Arithmeticam Jamblichi... . 
 Quarto. Also, (one publication, though different in place and 
 date, ) 
 
 Arnhem, sixteen-sixty-eight. Jamblichus. ‘In Nico- 
 machi Geraseni Arithmeticam introductionem et de Fato.’ 
 Quarto (Gr. Lat.). 
 
 These comments of Jamblichus and Camerarius are the easiest 
 accesses to a knowledge of the matter of the work of Nicomachus, 
 of whom no Latin version exists, of which I can find any mention. 
 
 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 Purchasers 
 Guide < for Builders 
 Mate Measurers. 
 
 Octavo (O). 
 
 The first book is on interest, the second and third on build- 
 ing and mensuration of work. 
 
 Toulouse, sixteen-seventy. Diophantus. ‘ Arithmeti- 
 corum libri sex.’ Folio in twos. 
 
 This is perhaps the best edition of Diophantus. It has the com- 
 
WORKS ON ARITHMETIC. 47 
 
 mentaries of Bachet and of Fermat. The character of the work of 
 Diophantus on the properties of numbers, treated algebraically, is 
 too well known to need description. I do not consider him as be- 
 longing to the genuine Greek school. His date is always spoken 
 of as very uncertain, though the preponderance of opinion seems 
 to place him in the second century. But I think 1 have given 
 sufficient reason for supposing him to have written as late as the 
 beginning of the seventh century. (See Dr. Smith’s Dictionary of 
 Biography, article Hypsicles.) 
 
 Diophantus is usually placed in the second century, because 
 Suidas says that the celebrated Hypatia wrote a comment on the 
 astronomical table of Diophantus. ‘This was a common name: Fa- 
 bricius has collected upwards of twenty writers or philosophers who 
 bore it; and the arithmetician of whom I am speaking shews no 
 appearance* of having attended to astronomy. Diophantus men- 
 tions Hypsicles, a mathematician: now (unless this mention make 
 a second) there is only one Hypsicles in Greek literature, namely, 
 the author of the two last books of Euclid’s elements. Of Hyp- 
 sicles, Suidas says that he was a pupil of ‘ Isidore the philosopher.” 
 But in another place (and this passage, occurring at the word Syri- 
 anus, has been overlooked till now) Suidas quotes Damascius for his 
 information upon this Isidore. Now it is certain that the Isidore, 
 whose life John Damascius wrote, must have been a contemporary of 
 the Emperor Justinian, at the earliest: and his pupil, Hypsicles, can 
 hardly have written before the middle of the sixth century. Dio- 
 phantus, who makes mention of Hypsicles, may have flourished 
 towards the end of the sixth, or the beginning of the seventh cen- 
 tury. The silence of Proclus, Pappus, and Theon, as to both Hyp- 
 sicles and Diophantus, is a strong presumption of the latter two 
 having come after the former three. When Dr. Peacock (p. 397) 
 says that Theon was well acquainted with the writings of Dio- 
 phantus, I suppose it is upon the presumption that the father and 
 teacher could not have been ignorant of the author on whom the 
 daughter and pupil was said to have published a commentary. But, 
 this apart, I cannot imagine any reason to suppose that Theon had 
 ever seen the work of Diophantus. P. 404. 
 
 London, sixteen-seventy-three. Sam. Morland. ‘The 
 Description and Use of two Arithmetick Instruments, together 
 With a Short Treatise explaining and Demonstrating the Ordi- 
 nary Operations of Arithmetick, As likewise A Perpetual Al- 
 manac And several Useful Tables.’ Octavo (duodecimo size). 
 
 There is also the following second title-page (first in order of time) 
 which is all that some copies have: London, sixteen-seventy-two. 
 ‘A New, and most useful Instrument for Addition, and Subtrac- 
 tion of Pounds, Shillings, Pence, and Farthings. .. . By S. Morland.’ 
 
 * For this reason Fabricius wants to alter the words ‘‘a commentary upon of Dio- 
 phantus the astronomical table,” into ‘‘a commentary upon Diophantus and an astro- 
 nomical table.” 
 
4s A CHRONOLOGICAL LIST OF 
 
 A very miscellaneous book, embodying computation (N), some of 
 Euclid, tables for Easter, description of the calculating machine, &c. 
 
 London, sixteen-seventy-three. John Kersey. ‘ M* Win- 
 gate’s Arithmetick, containing a plain and familiar method,’ 
 &e. 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) (0). 
 
 A book of a decidedly commercial character, and with good force 
 of examples. 
 
 London, sixteen-seventy-four. John Mayne. ‘NSocius 
 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 In- 
 troduction 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 (0). 
 
 Not a very easy system. 
 
 London, sixteen-seventy-seven. Michael Dary. ‘ Inte- 
 rest epitomised, both compound and simple... . whereunto 
 is added, a Short Appendix For the Solution of Adfected Equa- 
 tions in Numbers by Approachment: Performed by Loga- 
 rithms.’ Quarto (octavo size) (N). 
 
 Of Dary, who was a gunner, then a tobacco-cutter, then a teacher, 
 &c., and who was the correspondent of Collins, J. Gregory, Newton, 
 &c., a good deal is to be found in the Macclesfield correspondence. 
 This tract contains the distinct announcement and use of a’principle 
 which is now well known, and of which Newton’s (or Briggs’s) 
 method of approximation is only a particular case. If an equation 
 be reduced to the form a=ox, ¢x being a known function of a, suc- 
 cessive formation of pa=b, ¢b=c, pc=d, &c., gives an approximation 
 
WORKS ON ARITHMETIC. 49 
 
 to a root of the equation, whenever the results do not increase 
 without limit. Dary saw and applied this, expediting the process 
 occasionally by using the mean of the two last results, instead of the 
 last result alone, to produce the next result. The book itself is quite 
 out of notice; and neither the ingenuity nor the importance of the 
 principle was appreciated by Dary’s contemporaries. 
 
 London, sixteen-seventy-eight. William Leybourn. 
 ‘ Arithmetick Vulgar, Decimal, Instrumental, Algebraical.’ Oc- 
 tavo. Fourth edition. (The four parts have separate titles 
 and paging.) (ON). 
 
 IT 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 Hep- 
 tatechnists.’ Quarto. 
 
 This book I have several times met with in lists, as one of 
 arithmetic, so that I here insert a true account of it. It is written 
 against human learning, and arithmetic among the rest, as being of 
 no use to ‘ open the seals of the book,” or to interpret the Bible. It 
 reminds me much, difference of age and manners apart, of some 
 books I have lately seen, of arithmetical examples, in which no 
 name is introduced except from the Bible: so that Judas Iscariot 
 and the devil may be mentioned, but Socrates or Milton may not. 
 Of arithmetic T. Lawson says: ‘Herein any member of Italian 
 Babilon with his Mass-Book, Mass for the Dead, Fabulous Legend : 
 any Mahumetan with his dreggy Alcoran ; any Flint-hearted Jew 
 with his Talmud, a mingle-mangle of Jewish Divine and Humane 
 matters; any Dead, Dry, Unfruitful Formalist, may grow Profound, 
 Exquisit, Nimble, yea, and though involved in the intricate windings 
 of Degeneration, out of the Royal state of Regeneration and Heavenly 
 Transformation may apprehend the Feates, Termes and Parts of this 
 Natural Art, as Digits, Articles mixt Numbers, Cyphers, Terniries, 
 Golden Rule direct, Golden rule reverse, a Cube, Phythagoras’s 
 Table, Algorism, &c. yet be Strangers to the Divine Exercise which 
 leads to Christ, the Lion of the Tribe of Judah, who alone opens the 
 Seals of the Book.” Attention is then directed to the number of 
 the Beast. 
 
 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. ‘ Ough- 
 
 F 
 
50 A CHRONOLOGICAL LIST OF 
 
 tredus explicatus, sive Commentarius in Ejus Clavem Mathe- 
 maticam.’ 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. ‘ Ma- 
 thematice 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. ‘Amphi- 
 thalami or the Accountants closet.’ Folio. 
 
 Book-keeping by double entry. This and the two last books on 
 the subject (Dafforne and Collins) I have bound together with the 
 Lex Mercatoria of Gerard Malynes and other books. Mr. M‘Cul- 
 loch has precisely the same collection, and gives them all as one 
 publication (Liter. Polit. Econ. p. 129-30). But they have different 
 printers, different dates, and different signatures: most being in 
 twos, and the rest simple folios ; and from Collins’s preface it seems 
 that his book was certainly a separate publication. I merely men- 
 tion this (having seen the like in other cases) to advert to the proba- 
 bility of it not having been unusual for booksellers to bind parcels 
 of books on the same subject together for sale. 
 
 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 Cyel. ‘ 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 Geo- 
 metrical Key: or the Gate of Equations unlocked:’ &c. Quarto. 
 
 Polyglott: Latin and English. An attempt to do without arith- 
 metic in the solution of equations, by means of construction. See 
 the correspondence of Baker and Collins in the Macclesfield corre- 
 spondence, beginning of vol. ii. 
 
WORKS ON ARITHMETIC. al 
 
 London, sixteen-eighty-five. Seth Partridge. ‘The de- 
 scription and use of an instrument called the double scale of 
 proportion.’ Octavo. 
 
 On the sliding rule. 
 
 London, sixteen-eighty-five. John Collins. ‘The doc- 
 trine of decimal arithmetick, simple interest, as also of com- 
 pound interest .... made publick by T. D.’ Octavo (small). 
 
 A clear and short work, posthumous. 
 
 Naples, sixteen-eighty-seven. AGzgid. Franc. de Got- 
 tignies, ‘Logistica Universalis.’ folio in twos. 
 
 The author is the man mentioned by Montucla (ii. 648) as 
 having claimed some of the discoveries of Cassini. He was a Jesuit, 
 and was professor at Rome. This work is the last of several, of 
 which it seems to contain the substance, amplified (Heilbronner, iz 
 verb.). Ihave never met with this book till very lately ; and, though 
 I have had no time as yet to examine it fully, I have been much 
 surprised to find it unnoticed by mathematical historians. In his 
 views of algebra, the author seems much before his time. His re- 
 cognition ofits existence at that time as an art only; ofits containing 
 principles and definitions which are not arithmetic, but perfectly 
 distinct; its definition as a science in which subtraction is univer- 
 salised by express convention; the avowed enlargement of mean- 
 ing of + and— &c. &c.—shew that the author had got more than 
 a glimpse of what was coming. ‘The book is a folio of 400 pages, 
 with much prolixity of expression, but vigorous efforts of thought. 
 
 London, sixteen-eighty-seven. William Fiunt. ‘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. ‘ Aca- 
 demia 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. 
 
 UNIVERSITY oF 
 ILLINOIS LIBRARY 
 
52 A CHRONOLOGICAL LIST OF 
 
 A polyglott, English and French, announcing the intention of 
 setting up a College in London. It gives general ideas on the 
 sciences in dialogue; and that which relates to arithmetic passes 
 between the tutor and a page of the court, who begins thus: ‘If 
 you would favour me now with some particular discourses on all the 
 necessarie sciences (which would make me capable of the Theoricall 
 part), I should esteeme myselfe very much obliged unto you for 
 that it would make me passe for a gallant wit in good companies of 
 men that love to discourse pertinently of things.” With this view, 
 my predecessor (in intention) gives two pages and a half of arith- 
 metical terms, and then the young gentleman says: “‘ Ther’s enough 
 for my Theory.” This book was dated from Paris the day before the 
 funeral of Charles I. of England: but it was dedicated to the Duke 
 of York, with a hope that his turn for literature might increase the 
 comfort of his great parents. 
 
 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 Loga- 
 rithms, 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) (QO). 
 
 That this book should have reached a second edition shews how 
 great an object it was to save a little calculation. The author simply 
 approximates to the fractional part of the root by first interpolation. 
 Thus his general formula would be : 
 
 b 
 are b) 24 ee 
 ( lee (a+1)"—a" 
 to help which he has a small table of squares and cubes, with first 
 differences. But he evidently seems to think he has got the ewact 
 square and cube roots: and he says: 
 Now Logarithms lowre your sail, 
 And Algebra give place, 
 
 For here is found, that ne’er doth fail, 
 A nearer way, to your disgrace. 
 
 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. 
 
WORKS ON ARITHMETIC, 53 
 
 The arithmetic (ON) Vulgar, Decimal, Instrumental, and Alge- 
 braical, is stated to be the first book: but algebra is afterwards pro- 
 moted 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. ‘ Ele- 
 mens des Mathematiques,’ &c. Duodecimo. Third edition (QO). 
 
 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. ‘ Heptar- 
 chia 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 ge- 
 nerale des Quarrez magiques de quatre de costé.’ And in 
 another colleetion 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 Rect- 
 angles 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]. 
 
 Frenicle (who died in 1675) was one of those useless men, as many 
 appliers of mathematics have called them, who work subjects into 
 shape before the time is come for applying them. In this point we 
 have degenerated from our ancestors in the soundness of our views 
 as to what knowledge is, and how it comes. For instance, at the 
 end of the seventeenth century, Picard, an eminently useful (as 
 well as useful) applier of mathematics, who never speculated on 
 any subject, was the careful preserver of Frenicle’s manuscript 
 combinations, and exclusions, and magic squares. . A century after, 
 
 F 2 
 
54 A CHRONOLOGICAL LIST OF 
 
 Condorcet, a useful (but wseless) speculator, who never handled any 
 so-called practical subject, finds it necessary, in his éloge of Frenicle, 
 to apologise for the useless character of his writings, though the doc- 
 trine of probabilities,—our power over which depends on the know- 
 ledge of combinations, &c., in which Frenicle was a successful work- 
 man—was, when the éloge was written, almost as far advanced as the 
 theory of probable errors and the method of least squares, on which 
 every observer who knows his business now relies, and to which 
 astronomy in particular is much indebted for its modern accuracy. 
 Condorcet had caught a slang which was current in his day, and has 
 been very popular among us. The time is coming when really 
 learned men will again be ashamed of not seeing the value of all the 
 uses of mind: when nothing but thoughtlessness or impudence, 
 mercurial brain or brazen forehead, will aver that no knowledge is 
 practical, except that which ends in the use of material instruments. 
 
 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 owvrage trés estimable. He speaks of many 
 inferior works with much higher praise. 
 
 London, sixteen-ninety-four. Wm. Leybourn. ‘ Plea- 
 sure 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 fol- 
 lowing :—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). Wen- 
 terus Mandey. ‘Synopsis Mathematica Universalis: or the 
 Universal Mathematical Synopsis of John James Heinlin, Pre- 
 late of Bebenhusan.’ Octavo. 
 
 Translated from the third edition, Latin, Tubingen, sixteen- 
 ninety. Heinlin’s system of arithmetic (O) is somewhat theological. 
 ‘In Unity all numbers are virtually, although they are infinite. So all 
 
WORKS ON ARITHMETIC. 53 
 
 things are in God, in him they live and move. Seven is a Sacred 
 Number, chiefly used in Holy Scripture. It seems to have its Ori- 
 ginal from the Inscrutable Unity of Divine Essence, and Sextuple in 
 respect of the Divine Persons among themselves: whence also the 
 
 Description of a Septangled form is impossible, and cannot be known 
 by Human Minds.’ 
 
 London, sixteen-ninety-six. Samuel Jeake. ‘ Aoyiorixy- 
 Aoyia, 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 con- 
 tains, 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 con- 
 tagion, 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 cir- 
 cumstance 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 in- 
 tegers 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 unar- 
 ranged 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. 
 
 Ozford, sixteen-ninety-eight. E. Wells. ‘Elementa 
 arithmeticee numerose et speciosee.’ Quarto (octavo size) (N). 
 
 This is an elegant work, in which arithmetic and algebra are 
 exhibited to the university student in a simple form, accompanied 
 by the historical learning which such a student ought to have. The 
 works on arithmetic were taking too commercial a turn to be all- 
 sufficient for the purposes of liberal education: and Wells, without 
 by any means rejecting commercial questions, writes in order that 
 the student may not of necessity be driven to the works in which, 
 
56 A CHRONOLOGICAL LIST OF 
 
 as he says, evempla non aliunde petuntur quam a Butyro et Caseo, 
 Zingibere et Pipere, altisque consimilibus. 
 
 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 ‘‘ Arithmeticee 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 Incom- 
 parable Art, as it is now taught by the ablest School-Masters 
 in City and Country. Composed By Edward Cocker, late 
 Practitioner in the Arts of Writing Arithmetick, and En- 
 graving. 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 Mathe- 
 maticians and Writing-Masters in and near London.’ Duo- 
 decimo. 
 
 This is called ‘‘The twentieth edition carefully corrected, with 
 additions:”” but I find that it had become usual, when any one 
 edition was augmented, to make the assertion with reference to all 
 succeeding editions. The first edition of this famous work was in 
 sixteen-seventy-seven (I have seen one copy, which appeared in a 
 sale a few years ago), the fourth in sixteen-eighty-two, the twentieth 
 as above, the thirty-third in seventeen-fifteen, the thirty-fifth in 
 seventeen-eighteen, the thirty-seventh in seventeen-twenty. The 
 earliest edition I ever possessed is one of sixteen-eighty-five ; what 
 edition it is, is not stated. But there is confusion among the title- 
 pages. For though the above is unmistakeably marked seventeen- 
 hundred, and twentieth edition, I have also compared together two 
 editions, both of seventeen-twelve, one called the fourteenth, and the 
 other the thirtieth: all three by one printer, Eben. Tracy, at the 
 three Bibles on London Bridge. How long it went on in England 
 I do not know: but there was an edition at Edinburgh in seventeen- 
 sixty-five, and another at Glasgow in seventeen-seventy-one, both 
 edited by John Mair, whose preface is dated seventeen-fifty-one. 
 Some account of Cocker is given in the Penny Cyclopedia (art. 
 
WORKS ON ARITHMETIC. 57 
 
 ‘ Cocker’). Not much more is known of him than this, that he was 
 a skilful writing-master and engraver. At the beginning of the 
 Arithmetic is a recommendation signed ‘ John Collens’ (no doubt 
 the famous John Collins, or intended to pass for him) certifying that 
 the deceased author was ‘knowing and studious in the Mysteries 
 of Numbers and Algebra, of which he had some choice Manu- 
 scripts and a great collection of Printed Authors in several Lan- 
 guages.”’ Collins doubts not “but he hath writ his Arithmetick 
 suitable to his own Preface and worthy acceptation :’”’ which means 
 that Collins or Collens had only seen the preface of the forthcoming 
 work at most. Then follows the attestation of fifteen teachers to the 
 merits of the work. All this looks odd; because, according to the 
 editor, the book was that which had been long promised to the world 
 by a celebrated writer. All attestation was unnecessary; and the 
 certificate of a celebrated name, wrong spelt, to the effect that he 
 had no doubt the work, then printed, would be good, may now ex- 
 cite a little curiosity, if not suspicion. 
 
 I am perfectly satisfied that Cocker’s Arithmetic is a forgery 
 of Hawkins, with some assistance, it may be, from Cocker’s papers : 
 that is to say, there has certainly been more or less of forgery, with- 
 out any evidence being left as to whether it was more or less. I 
 could easily believe that all was forged; and my reasons are as 
 follows : 
 
 In both the editions of Hodder which I have seen (1664 and 
 1672) is the following advertisement: ‘‘ There is newly printed Mr. 
 Cocker’s book called the Tutor to Writing and Arithmetic.”’ It ap- 
 pears then that during his lifetime he had published a book on Arith- 
 metic, which I suspect to have been what would now be called an 
 arithmetical copy-book, with engraved questions and space left for 
 the work. But neither the posthumous work, nor its preface signed 
 by Cocker himself, make the least allusion either to the previous 
 work, or to the promise of another. On the contrary, the lan- 
 guage of Cocker’s own preface implies that it is the first work he 
 has published on arithmetic, and agrees with many other prefaces 
 (which are usually written last) in speaking of the work as already 
 published. ‘To establish these and other contradictions, I first give 
 Hawkins’s account in his preface (with my own Italics). ‘‘ I Having 
 the Happiness of an Intimate Acquaintance with Mr. Cocker in his 
 Life-time often solicited him to remember his Promise to the World, 
 of Publishing his Arithmetick, but (for Reasons best known to him- 
 self) he refused it; and (after his Death) the Copy falling accident- 
 ally into my hands, I thought it not convenient to smother a work 
 of so considerable a moment,” &c. But Cocker himself writes, or 
 is made to write, as follows: ‘ By the sacred Influence of Divine 
 Providence, I have been Instrumental to the benefit of many ; by 
 vertue of those useful Arts, Writing and Engraving: and do now 
 with the same wonted alacrity cast this my Arithmetical Mite into 
 the Publick Treasury. . . . For youthe pretended Numerists* .. . 
 
 * Numerist is a wordwhich Hawkins uses in his own professed writings: and it 
 was by no means a common word. 
 
58 A CHRONOLOGICAL LIST OF 
 
 was this Book composed and published. .. .’’ This is an odd pre- 
 face for a book which the author never meant to publish, and re- 
 fused to publish, though pressed to do so. Of course it is possible 
 that though he wrote with an intention of publishing, he afterwards 
 changed his mind. This is one explanation; that Hawkins forged 
 clumsily is another: and which is the most probable must be 
 gathered from a review of all the circumstances. 
 
 Next, at the end of the work, Hawkins gives a hint of a book on 
 decimals which would be forthcoming in time. Accordingly, we 
 have: 
 
 London, sixteen-eighty-five. ‘Cocker’s Decimal Arithmetick .. . 
 Whereunto is added his Artificial Arithmetick . . . . also his Alge- 
 braical 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. 
 
 This book came toits third edition in seventeen-two. The artificial 
 (or logarithmic) arithmetic, and the algebra, have separate title- 
 pages, dated sixteen-eighty-four. Cocker gives no preface here, 
 but Hawkins does, stating that he had in the preceding work given 
 ‘‘an account of the speedy publication of his Decimal, Logarithmical, 
 and Algebraical Arithmetick.’’ He has here mended his hand : for, 
 except the words ‘such Questions being more applicable to Deci- 
 mals are omitted till we come to acquaint the Learner therewith,”’ 
 the first treatise does not give a hint ofthe second. Again, Kersey’s 
 Algebra, on which part of Cocker’s second work is founded, was 
 published in 1673, and the latter had been dead some time before 
 the manuscript of the first work of 1677 “accidentally” fell into 
 Hawkins’s hands. This is again singular, on any supposition but 
 that of the forgery. Moreover, at the end of the preface, Hawkins 
 writes a letter to his friend John Perkes, in cipher (Penny Cyel. 
 ‘ Cocker’) in which he says, ‘‘ If you peleas to bestow some of your 
 spare houres in perusing the following tereatise, you will then be 
 the better able to judg how I have spent mine.” This looks like a 
 confession of authorship. And in 1704, as presently noted, appeared 
 Cocker’s English Dictionary by John Hawkins, who would perhaps, 
 had he lived, have found Cocker’s Complete Dancing-Master and 
 Cookery-book among the papers of the deceased. 
 
 The famous book itself I take to be a compilation or close imi- 
 tation in allits parts. Even the Frontispiece, &c. is fashioned upon 
 Hodder. ‘Thus, Hodder begins with his own portrait, and verses 
 of exaggerated praise under it; and so does Cocker. The former 
 begins his title with Hodder’s Arithmetick, the latter with Cocker’s 
 Arithmetic. ‘The former speaks of that necessary art, the latter of 
 that incomparable Art. ‘The former has it ‘explained in a way 
 familiar to the capacity,’ the latter a ‘familiar method suitable to 
 the meanest capacity :’ the two words being then by no means so 
 common in the senses put upon them as they are now. Turning 
 over the title-page we find that each of them ‘humbly dedicateth 
 this Manual (manuel in Hodder) of Arithmetick,’ the first to a 
 
WORKS ON ARITHMETIC, 59 
 
 ‘most worthily honoured friend,’ the second to ‘much honoured 
 friends ;’ the first ‘in token of true gratitude for wnmerited kind- 
 nesses,’ the second ‘as an acknowledgment of wnmerited favours.’ 
 There are too many small coincidences here. And it must be re- 
 membered that every resemblance to a work so well known as 
 Hodder’s (it is one of the few English works of the century which 
 have found their way into Heilbronner’s list) would help the 
 sale. 
 
 From all these circumstances I was tolerably sure that there was 
 no dependence to be placed on the famous Cocker being any body 
 but Hawkins, so far as this book is concerned: though I must say 
 I hardly expected to find such confirmation as would arise from 
 catching Hawkins at a similar trick in another quarter. But on 
 looking at the work above described as the third edition of Jonas 
 Moore’s Arithmetick, my eye was caught by the following sen- 
 tence: ‘You may likewise prove Division by Division, as [ have 
 shewed at large in the 7. chap. Page 100, 101, 102. of Mr. 
 Cocker’s Arithmetic, printed in the year 1685.” Now Jonas 
 Moore was dead before 1685; and moreover, could have shewn 
 nothing in Cocker’s Arithmetic: and on looking farther to see who 
 it is that thus speaks in the first person, I find the name of John 
 Hawkins to the second part of the work, as editor, And on looking 
 farther, I find that a good deal from Moore’s own editions has been 
 introduced verbatim into Cocker. For instance, this sentence is in 
 both: ‘* Notation teacheth how to describe any number by certain 
 notes and [or] characters, and to declare the value thereof being so 
 described.” And throughout the book, paragraphs are frequently 
 introduced from Moore, with alterations of phrase here and there. 
 So that we have Hawkins arbitrarily altering and adding, in the 
 first person, to the text of a book which had been for thirty years 
 before the world under Moore’s name. What are we to suppose 
 he would do with Cocker’s papers, if indeed he had any? More- 
 over, we find in Cocker sentences which had been previously written 
 by Moore. 
 
 To see whether much was gained by Cocker’s Arithmetic, as 
 well as for the interest of the comparison itself, I will write down 
 the definitions of addition, subtraction, multiplication, and division, 
 from Recorde, Wingate, Johnson (see Additions to ee work), Moore, 
 Bridges, Hodder, and Cocker. 
 
 ADDITION. 
 
 Recorde. Addition is the reduction and bringing of two summes 
 or more into one. 
 
 Wingate. Addition is that by which divers Numbers are added 
 together, to the end that their sum, aggregate, or total, may be dis- 
 covered. 
 
 Johnson. Addition serveth to adde or collect divers summes of 
 severall denominations, and to expresse their totall value in one 
 summe. 
 
 Moore. Addition is that part of Numbring or Numeration, 
 
60 A CHRONOLOGICAL LIST OF 
 
 whereby two or more numbers are added together, and so the totall 
 or summe of them is formed. 
 
 Bridges. Addition is the gathering together and bringing of 
 two numbers or more into one summe. 
 
 Hodder. Addition teacheth you to add two or more sums to- 
 gether, to make them one whole or total sum. 
 
 Cocker. Addition is the Reduction of two, or more numbers of 
 like kind together into one Sum or Total. Or it is that by which 
 divers numbers are added together, to the end that the Sum or Total 
 value of them all may be discovered. 
 
 SUBTRACTION. 
 
 Recorde. Subtraction diminisheth a grosse sum by withdrawing 
 of other from it, so that Subtraction or Rebating is nothing els, but 
 an arte to withdrawe and abate one sum from another, that the Re- 
 mainer may appeare. 
 
 Wingate. Subtraction is that by which one number is taken out 
 of another, to the end that the remainder, or difference, between the 
 two numbers given may be known. 
 
 Johnson. Subtraction serveth to deduct one summe from an- 
 other ; the lesser from the greater, and to shew the remaines. 
 
 Moore. Substraction is that part of Numeration where one num- 
 ber is substracted or taken out of another, and so the Remainder is 
 gotten, which is also called the difference or excesse. 
 
 Bridges. Subtraction is the taking of one number from another, 
 whereby the residue, remainder or difference is found. 
 
 Hodder. Substraction teacheth to take any lesser number out of 
 a greater, and to know what remains. 
 
 Cocker. Subtraction is the taking of a lesser number out of a 
 greater of like kind, whereby to find out a third number, being or 
 declaring the Inequality, excess, or difference between the numbers 
 given, or Subtraction is that by which one number is taken out of 
 another number given, to the end that the residue, or remainder 
 may be known, which remainder is also called the rest, Remainder, 
 or difference of the numbers given. 
 
 MULTIPLICATION. 
 
 Recorde. ‘Multiplication is such an operation, that by two summes 
 producyth the thirde: whiche thirde summe so manye tymes shall 
 containe the fyrst, as there are unites in the second. And it ser- 
 veth in the steede of many Additions. 
 
 Wingate. Multiplication teacheth how by two numbers given to 
 find a third, which shall contain either of the numbers given, so 
 many times as the other contains 1 or unitie. 
 
 Johnson. Multiplication is a number of additions speedily per- 
 formed. 
 
 Moore. Multiplication, is a part of conjunct Numeration, or num- 
 bring, whereby the Multiplicand (which is the number to be multi- 
 plied) is so often added to it selfe, as an unite is contained in the 
 
WORKS ON ARITHMETIC, 61 
 
 Multiplyer (which is the number multiplying) and so the Factus (or 
 Product) which is the result of the worke, is had. 
 
 Bridges. Multiplication (which serveth for many additions) is 
 that by which we multiply two numbers the one by the other, to the 
 end their product may be discovered. 
 
 Hodder. Multiplication serveth instead of many additions, and 
 teacheth of two numbers given to increase the greater as often as 
 there are Unites in the lesser. 
 
 Cocker. Multiplication is performed by two numbers of like 
 kind, for the production of a third, which shall have such reason to 
 the one, as the other hath to unite, and in effect is a most breif 
 and artificial compound Addition of many equal numbers of like 
 kind into one sum. Or Multiplication is that by which we multiply 
 two or more numbers, the one into the other, to the end that their 
 Product may come forth, or be discovered. Or, Multiplication is 
 the increasing of any one number by another; so often as there 
 are Units in that number, by which the other is increased, or by 
 having two numbers given to find a third, which shall contain one 
 of the numbers as many times as there are Units in the other. 
 
 DIVISION. 
 
 Recorde. Division is a partition of a greater summe by a lesser. 
 
 Wingate. Division is that by which we discover how often one 
 number is contained in another, or (which is the same) it sheweth 
 how to divide a number propounded into as many equal parts as 
 you please. 
 
 Johnson. Apparently considers division not enough of a techni- 
 cal term to need definition: his first example is, ‘‘I would divide 
 65490 pound amongst 5 men.”’ 
 
 Moore. Division is that part of conjunct Numeration, wherby 
 one Number is substracted from another, as often as it is contained 
 in it, and by that meanes it is found how many of the one is cofi- 
 tained in the other. Baa 
 
 Bridges. Division is that by which we discover how often one 
 number is contained in another. 4 
 
 Hodder. Division is that by which we know how many times a 
 lesser sum is contained in a greater. 
 
 Cocker. Division is the Separation, or Parting of any Number, or 
 Quantity given, into any parts assigned; Or to find how often one 
 Number is Contained in another; Or from any two Numbers given 
 to find a third that shall consist of so many Units, as the one of 
 those two given Numbers is Comprehended or contained in the other. 
 
 The six predecessors of Cocker whom I have chosen, stop when 
 they think enough has been said. But the illustrious discoverer, 
 or at least the first general propagator, of the fact that two and 
 two make four (for his current reputation amounts to this) must 
 have had more in view. He seems to be laying every offence against 
 accuracy in different ways, so that the unfortunate schoolboy who 
 commits it may be sure of a flogging under one count or another of 
 
 G 
 
62 A CHRONOLOGICAL LIST OF 
 
 his definition. And the vice of confounding abstract and concrete 
 number, which leads him to imply that five shillings can be multi- 
 plied by five shillings, runs through his whole book: as does also 
 the tendency to prolixity and reduplication of things which confuse 
 each other. As to the general notion of what Arithmetic is, Cocker 
 tells his beginners that it is either Natural, Artificial, Analytical, Al- 
 gebraical, Lineal, or Instrumental. The natural is ‘that which is 
 performed by the Numbers themselves; and this is either Positive or 
 Negative. Positive, which is wrought by certain infallible numbers 
 propounded, and this either Single or Comparative; Single, which 
 considereth the nature of numbers simply by themselves ; and Com- 
 parative, which is wrought by numbers as they have Relation one to 
 another. And the Negative part relates to the Rule of False.’ Arti- 
 ficial Arithmetic is performed by artificial or borrowed numbers in- 
 vented for that purpose, called Logarithms. Analytical Arithmetic 
 ‘is that which shews from a thing unknown to find truly that which 
 is sought; always keeping the Species without Change.’’ Alge- 
 braical Arithmetic ‘‘ is an obscure and hidden Art of accompting by 
 numbers in resolving of hard Questions.’’ Lineal Arithmetic “is that 
 which is performed by lines, fitted to proportions, as also Geometrical 
 projections.”” Instrumental Arithmetic “is that which is Performed 
 by Instruments, fitted with Circular and Right lines of proportions, 
 by the motion of an index or otherwise.’’ So much for Cocker (or 
 Hawkins) as an explainer. As to the actual modes of operation, they 
 are neither better nor worse than those pointed out before by Wingate, 
 Moore, and Bridges. The famous book looks like a patchwork col- 
 lection, and, I believe, is nothing more. The reason of its reputa- 
 tion I take to be the intrinsic goodness of the processes, in which 
 the book has nothing original; and the systematic puffing with 
 which it was introduced. The long-promised book of the great Mr. 
 Cocker, with Collins and fifteen other teachers to recommend it, 
 pushed aside better productions. I am of opinion that a very great 
 deterioriation in elementary works on arithmetic is to be traced 
 from the time at which the book called after Cocker began to pre- 
 vail. This same Edward Cocker must have had great reputation, 
 since a bad book under his name pushed out the good ones. 
 
 London, seventeen-hundred. Christopher Sturmius. 
 ‘Mathesis Enucleata, or the Elements of the Mathematicks. 
 .... Made English by J. R. M. and R. 8. 8.’ Octavo. 
 
 Arithmetic and Algebra are here very nearly separated. 
 
 Edinburgh, seventeen-one. George Brown. ‘ A Com- 
 pendious, 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. 
 
WORKS ON ARITHMETIC, 63 
 
 . London, seventeen-three. John Parsons and Thos. 
 Wastell. ‘Clavis Arithmeticee.’ 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. ‘ Arith- 
 meticz 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 Mathe- 
 matics.’ 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 demonstrata. Cui 
 accesserunt, Cogitata nonnulla de Radicibus Surdis, de stu 
 Aeris, de Ludo Algebraico, &e,’ Octavo in twos (NO). 
 
 There is no doubt that the author was the celebrated Bishop Ber- 
 keley, 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). 
 
 This is the first edition of these celebrated tables, and is little 
 known. The second edition, London, seventeen-twenty-six, quarto, is 
 the best having Smart’s name. But in reality the Tables in Francis 
 Baily’s ‘ Doctrine of Interest and Annuities,’ London, eighteen-eight, 
 quarto, are Smart’s, and do not profess to be any thing else. Mr. 
 Baily (who, by the way, did not know this first edition) says, ‘ I 
 have neither time nor inclination to calculate them anew; and there- 
 fore I give them to the world with all their imperfections on their 
 head. Iam happy, however, to observe, that after many years ex- 
 perience, I have not met with any errors but such as might be dis- 
 covered on inspection.’ Brand’s edition of seventeen-eighty is said 
 by Mr. Baily to have a good many errors. 
 
 There is yet another edition, London, seventeen-thirty-six. John 
 Smart. ‘Tables of Interest, &c. Abridged for the Use of Schools, 
 in Order to Instruct Young Gentlemen in Decimal Fractions.’ Oc- 
 tavo. Here is another instance of what I have before remarked, that 
 compound interest has always been considered the application of de- 
 cimal fractions, by those whose arithmetic has been commercial. 
 
64 A CHRONOLOGICAL LIST OF 
 
 Manuscript (in my possession) no place, after seventeen-ten. 
 
 I insert this here, as proving that, so late as the date above 
 mentioned, there were French schools in which the decimal point 
 was not introduced, the old method of division was employed, and 
 the Ptolemaic system was taught. It is the collection of notes of lec- 
 tures made by a young Englishman educated in France, and, was 
 sold a few years ago by his descendant, 
 
 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 gen- 
 tleman’s,’ on the other ‘arithmetic,’ ‘geometry,’ or ‘mechanics,’ 
 as the case may be. The gentlemen. are those whom God has re- 
 lieved 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 Under- 
 lings as Shop-keepers or Trades-men ;’ 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.’ Duo- 
 decimo. ‘Twelfth edition; with an Appendix on Book-keeping 
 by Chas. Snell. 
 
 A work of the immediate school of Cocker. 
 
WORKS ON ARITHMETIC. 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 be- 
 lieve, 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, 
 
 No place marked, seventeen-%". George Brown. 
 ‘ Arithmetica Infinita, or the Accurate Accomptant’s Best’ Com- 
 panion.’ 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, 1d. are given the multiples of :20520833..... The 
 whole work is copperplate engraving from beginning to end. From 
 several indications, ] gather that Geo. Brown of this work is also the 
 author mentioned under seventeen-one. : 
 
 London, seventeen-seventeen, Roger Rea. ‘The Sec- 
 tor 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 Trigo- 
 nometry.’ Octavo (N). 
 
 The treatise of an illiterate and confused person. Nothing has 
 been more common than for those who write on application to consi- 
 der 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 com- 
 monly 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). 
 G2 
 
66 A CHRONOLOGICAL LIST OF 
 
 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. Una 
 cum totidem Arithmetices Chronologice Libellis.’ Third edi- 
 tion. 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 Mer- 
 chant’s Magazine : or, Trades-Man’s Treasury.’ Quarto. 
 
 This is the eighth edition of a work of some celebrity, but which . 
 must not be confounded with Hatton’s edition of Recorde. The 
 only guide to the date of the first edition which I have is the state- 
 ment of the eighth that it was reviewed in the Ouvrages des Savans* 
 for 1695. 
 
 It is interspersed with copperplate pages of flourished writing, 
 containing examples and definitions. ‘There is somewhat more of 
 reason given for rules than was very common, and a vast quantity of 
 mercantile terms, usages, &c. are explained. 
 
 Witemberg, seventeen-twenty-seven. Joh. Fr. Weidler. 
 ‘De Characteribus Numerorum Vulgaribus et eorum Atatibus 
 . . « . Dissertatio Critico-mathematica.’ Quarto. 
 
 London, seventeen-twenty-eight. E. Hatton. ‘A Ma- 
 thematical 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 
 affairs of life and commerce being also fully explained: so as to 
 
 * Or else the Acta Eruditorum. This comes of translating. The phrase is “works 
 of the learned.” 
 
WORKS ON ARITHMETIC. 67 
 
 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 In- 
 tire System of Arithmetic... . containing, I. Vulgar. II. De- 
 cimal. III. Duodecimal. IV. Sexagesimal. V. Political. VI. 
 Logarithmical. VII. Lineal. VIII. Instrumental. IX. Alge- 
 braical. 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 me- 
 thod on any rule: but the first example taken at hazard will pro- 
 bably defy the abbreviations. 
 
 London, seventeen-thirty-two. Joseph Champion. ‘ Prac- 
 tical Arithmetick compleat.’? Octavo in twos. 
 
 London, seventeen-thirty-five. John Kirkby. ‘ Arithme- 
 tical 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 extrac- 
 tion 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 Arith- 
 THEUC a fines esills > 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 copper- 
 plate, 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. 
 
68 A. CHRONOLOGICAL LIST OF 
 
 London, seventeen-thirty-five. Benjamin Martin. ‘A 
 new Compleat and Universal System or Body of Decimal Arith- 
 metick.’ 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. 
 
 © Arithmeticee et 
 
 _ Edinburgh, seventeen-thirty-six. 
 Algebree 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 
 tn 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. ‘ Stereo- 
 metry .... . by the Help of a Sliding-Rule.’ Edited by Lead- 
 better, tenth edition. Duodecimo. 
 
 A book which once had a great reputation among excise col- 
 lectors. 
 
 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 pro- 
 perties of numbers. 
 
 Cambridge, seventeen-forty. Nicholas Saunderson. 
 ‘The Elements of Algebra, in ten books.’ Quarto, two vo- 
 lumes. 
 
 This is a posthumous work of the well-known Professor Saun- 
 derson, the blind lecturer on uptics. 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. ‘ Arith- 
 metick in Epitome,’ Sixth edition. Duodecimo (ON). 
 
 The eighth edition of the author’s book-keeping is London, seven- 
 
WORKS ON ARITHMETIC,» 69 
 
 teen-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 con- 
 tinued 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 Arith- 
 metick ; where, by the Bait of low Prices, he perhaps gathers a Num- 
 ber 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 fatut 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 prede- 
 cessors 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. Heilbron- 
 ner. ‘Historia Matheseos Universe.’ Quarto. 
 Though called a history of mathematics, and really a:biblio- 
 
 graphy raisonée, yet it is peculiarly devoted to arithmetic, the authors 
 on which have a separate list. There are also dissertations on nu- 
 
70 A- CHRONOLOGICAL LIST OF 
 
 merals, on their history, &c. The index of this book is of rare 
 goodness. 
 
 Geneva, seventeen-forty-three, forty-six, forty-seven, forty- 
 nine, forty-one. Christian Wolff. ‘Elementa Matheseos 
 Universe.’ Quarto. A second or later edition (N). Five 
 volumes. 
 
 The first of the five volumes contains a short treatise on arith- 
 metic. 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 whichis 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 Hum- 
 phrey 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.’ , Oc- 
 tavo 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.’ Duo- 
 decimo in threes. 
 
 This is a work both learned and foolish: but with the learning 
 and folly so distinct that they can be used separately. The folly 
 consists mostly in an attempt to give the rules of Arithmetic in 
 English hexameters and in alphabetical order; I give a couple of 
 instances. 
 
WORKS ON ARITHMETIC. a 
 
 Barter. 
 
 Barter, exchange of commodities: the rule to proportion ’em as follows : 
 
 What’s to be changd, Value: then, see what That will purchase of T’other. 
 
 If an advanc’d price of one, a proportionable find for the other. 
 
 Casting out of Nines. 
 
 Prove by a careful review: ’tis the safest: the readiest, as follows: 
 
 Sus.] right; when -hend and remainder (together) make up the compound. 
 
 App Mutt Dtv] add the digits together and cast-out the nines: then 
 
 Right ; if remainder of Facits agrees with remainder of factors, 
 
 Multiplied in Mul: -sor and quotient in Div; to which add the remainder. 
 
 The learning consists in a great knowledge of former writers, 
 
 and a copious account of weights, measures, and coins: together 
 with a list of authors, which I have copied into my own, so far as 
 I could not find the names elsewhere. 
 
 London, seventeen-fifty. James Dodson. ‘The Ac- 
 countant, 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 Antiloga- 
 rithmic 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 under- 
 stand 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 prac- 
 tice. 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. David- 
 son. ‘The Practical Arithmetician: or Art of Numbers im- 
 proved.’ 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 Simpson in 
 
72 A CHRONOLOGICAL LIST OF 
 
 his place. But 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 con- 
 vey an indecent double-meaning, in a manner obviously intended. 
 From 1750 or thereabouts, he began to write against Thomas Simp- 
 son. One of his publications against the latter is headed, ‘‘ Miss 
 Billingsgate in a salivation for a black eye:’’ and in a letter pub- 
 lished 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 ques- 
 tion 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 edi- 
 tion. 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 ear- 
 liest of the books which are now called ready reckoners, meaning 
 those which have totals at given prices ready cast up, was the fol- 
 lowing: London, sixteen-ninety-three. Wm. Leybourn. ‘ Pan- 
 arithmologia; 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). 
 
 This good book fulfils to a great extent the profession of the title- 
 
WORKS ON ARITHMETIC. 73 
 
 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 Mathema- 
 tical 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 biogra- 
 
 phy), 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 decimais. 
 
 London, seventeen-sixty. Jacob Welsh. ‘The School- 
 master’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 Prac- 
 tice 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. ‘Arith- 
 metica Universalis.’ Quarto. 
 
 This is the edition published by Castiglione, in two volumes, and 
 is the best. The original book, published by William Whiston, con- 
 sisted of the records laid up in the University Archives of the lec- 
 tures which Newton delivered as Lucasian professor. [S’ Graves- 
 ande, in the preface to his edition of Leyden, seventeen-thirty-two, 
 says it was published without the author’s knowledge,and much to 
 his displeasure :. but Whiston, in his Memoirs, says it was with New- 
 ton’s consent: most likely it was both with his consent, and to his 
 displeasure. Though properly a book on Algebra, and its appli- 
 cation to Geometry, yet it does contain a system of Arithmetic. 
 Various alterations, both of matter and arrangement, were made in 
 
 H 
 
74 A CHRONOLOGICAL LIST OF 
 
 the (so called) second edition [By Machin, 1722], which are sup- 
 posed to have been approved, if not furnished, by Newton himself. | 
 This information comes from Castiglione. The other editions 
 I have seen are the original Latin of Whiston, Cambridge, seven- 
 teen-seven, octavo. London, seventeen-twenty. ‘ Universal Arith- 
 metic,.... translated ....by the late Mr. Raphson, and revised 
 and corrected by Mr. Cunn.’ Octavo in twos. The same reprinted, 
 London, seventeen-twenty-eight, octavo in twos, called second edi- 
 tion; it is advertised as carefully compared with the correct edition 
 that was published in seventeen-twenty-two. There is also Wilder’s 
 edition of Raphson’s edition, London, seventeen-sixty-nine, octavo. 
 
 London, seventeen-sixty-five. J. Randall. ‘An Intro- 
 duction To so much of the Arts and Sciences, More immedi- 
 ately concerned in an Excellent Education for Trade In its 
 lower Scenes and more genteel Professions,’ &e. 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 arrange- 
 ments, 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 Gentle- 
 man 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. ‘ Mer- 
 cantile 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 At- 
 tempt to illustrate the Usefulness of Decimal Arithmetic.’ 
 Second edition. Octavo (small). 
 
 This book contains what I thought no one had given before my- 
 
 self, a complete head-rule for turning fractions of 1/. into decimals 
 to any number of places. This method, which is much wanted in 
 
WORKS ON ARITHMETIC. 75 
 
 commercial arithmetic, is here lost amidst attempts to compute with 
 interminable fractions; things on which real business will never 
 waste a thought. 
 
 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 privilége 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 Baréme,’ Paris, seventeen-ninety-eight, octavo, which he 
 called a Baréme décimal, in which there is a well-marked distinction 
 between Baréme the person and Baréme 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 1/. 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 2 of 180, namely 108; then 108 is 
 divided by 180, producing °6; then each of the parcels is multi- 
 plied 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 currous WORK, Totally 
 different from all that have preceded it.’ Octavo in twos. 
 
 There is something new, says the author, in almost every rule: 
 
76 A CHRONOLOGICAL LIST OF 
 
 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 loga- 
 rithms, Xe. 
 
 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. 
 
 Ozford, 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 in- 
 quirers into mathematical history, has given an account of Ander- 
 son, 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 com- 
 mendations prefixed, it would seem that the first edition was pub- 
 lished in seventeen-forty-four or forty-five. 
 
 Great-Yarmouth, seventeen-eighty-five. Thos. Sutton. 
 
WORKS ON ARITHMETIC. ra 
 
 ‘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 Compu- 
 tations which occur in the Practice of Numbers.’ Quarto. 
 
 Edinburgh, seventeen-eighty-six. John Mair. ‘ Arith- 
 metic, 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, seven- 
 teen-eighty-eight (the other). James Williamson. ‘The 
 Elements of Euclid, with dissertations... .’ Quarto. 
 
 The arithmetical books of Euclid are here with the rest. I have 
 chosen this edition by which to introduce the name, because it is 
 the only modern translation of Huclid. All the works which go by 
 that name are versions thickly scattered with the views of the editors 
 as to what Euclid ought to have been, instead of the rendering of 
 what he was. For these “‘ many tamperings with his text,’’ a coun- 
 tryman* of Robert Simson has been the first to call them the ‘ per- 
 
 * Sir William Hamilton of Edinburgh, in his notes to Reid, p. 765. 
 H 2 
 
78 A CHRONOLOGICAL LIST OF 
 
 fidious editors and translators of Euclid ;” a name which, in a sense, 
 they richly deserve. Williamson was a real disciple of Euclid; and 
 he translated so closely, that such words as, not being in the Greek, 
 English idiom renders necessary, are put in Italics. For many edi- 
 tions of Euclid, the reader may consult my article on that name, in 
 Dr. Smith’s Biographical Dictionary: and, for the contents of the 
 tenth book, the article Irrational Quantities in the Penny Cyclo- 
 edia. 
 
 4 If the demonstrative system of Euclid had taken as great a hold 
 in arithmetic as in geometry, we should not have had to complain of 
 one of the best exercises of thought being employed for no other 
 purpose than to make machines. 
 
 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. Duode- 
 cimo 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. ‘Cy- 
 clomathesis, 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. Emer- 
 son 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’ Arith- 
 métique raisonnée et démontrée, oeuvres posthumes de Léo- 
 nard Kuler, traduite en Francois par Bernoulli, Directeur de 
 V Observatoire de Berlin, &c. &c.’? Octavo in twos (ON). 
 
 The editor calls this the first work of Euler in his preface, and 
 
WORKS ON ARITHMETIC, 79 
 
 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 dis- 
 ciplines. 
 
 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 Lon- 
 don, 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 Systéme de Calcul. Third edition. 
 Octavo. 
 
 4 
 
 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. 
 
 This work, I am told, had such extensive currency in Ireland 
 
80 A CHRONOLOGICAL LIST OF 
 
 (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, after- 
 wards reduced again for schools. It is a book of the ordinary cha- 
 racter, 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 Mfilns. 
 ‘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 forma- 
 tion of its language. 
 
 London, seventeen-ninety-eight. Rich. Chappell. ‘The 
 Universal Arithmetic.’ Octavo in twos (small). 
 
 This book deserves notice for the author’s attempt to introduce 
 the practice of subtracting in division, without writing down the 
 subtrahend. He versifies his tables, ex. gr. 
 
 So 5 times 8 were 40 Scots 
 Who came from Aberdeen, 
 
 And 5 times 9 were 45,* 
 Which gave them all the spleen. 
 
 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 edi- 
 tion; 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 these are dwodecimo. When editors do not agree 
 
 * The North Briton, No. 45. 
 
WORKS ON ARITHMETIC, 81 
 
 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. ‘ Li- 
 neal 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.’ Duo- 
 decimo in sixes. (Tenth edition, edited by J. Warburton ; 
 eleventh in eighteen-one.) 
 
 In the same place, year, form, and by the same editor, was pub- 
 lished a new edition of the Key. It appears that the first edition 
 was reviewed in the Monthly Review for 1771, so that it is probably 
 of the year before. Vyse is one of the most celebrated of the illus- 
 trious band who used to adorn the shelves of a country schoolmaster 
 at the beginning of this century; Vyse, Dilworth, Walkingame, 
 Keith, Joyce, Hutton, Bonnycastle (with Cocker for a lost Pleiad). 
 He is also the poet of the lot: and some of his examples have gone 
 through many other books. The following specimen of the muse 
 of arithmetic should be preserved, as the best known in its day, and 
 the most classical of its kind : 
 
 When first the Marriage-Knot was tied 
 Between my Wife and me, 
 
 My Age did her’s as far exceed 
 As three Times three does three; 
 
 But when ten Years, and Half ten Years, 
 We Man and Wife had been, 
 
 Her Age came up as near to mine 
 As eight is to sixteen. 
 
 Now, tell me, I pray, 
 
 What were our Ages on the Wedding Day? 
 
 The book (this tenth edition at least) is crowded with examples, 
 which circumstance makes the Key very large. On the execution 
 there is no remark to make. Ifa new edition were published, some 
 
 of the examples must be omitted, as rather opposed to modern ideas 
 of decency. 
 
 Paris, eighteen-hundred. ‘Séances des Ecoles Normales, 
 recueillies par des sténographes, et revues par les Professeurs. 
 Nouvelle édition.’ Thirteen volumes. Octavo, 
 
 When the Normal School was founded at Paris, in 1794, the 
 professors engaged ‘“‘ pledged themselves to the representatives of 
 the people and to each other’ neither to read nor to repeat from 
 memory. ‘Their lectures were taken down in short-hand, and these 
 volumes contain some of them. The professors of mathematics were 
 Lagrange and Laplace: and few persons are aware that the mode in 
 
82 A CHRONOLOGICAL LIST OF 
 
 which the two first mathematicians in Europe taught the humblest 
 elements of arithmetic and algebra can thus be judged of. The 
 contents are, so far as these subjects are concerned; vol. I. p. 16, 
 programme and Laplace, arithmetic ; 268, Laplace, arithmetic ; 
 381, Laplace, algebra: II. 116, Laplace, algebra; 302, do. do.: III. 
 24, Laplace, algebra; 227, Lagrange, arithmetic; 276, Lagrange, 
 algebra; 463, do. do.: IV. 41, Laplace, geometry; 223, Laplace, 
 algebraic geometry ; 401, Lagrange, algebraic geometry: V. 201, 
 Laplace, new system of measures: VI. 32, Laplace, probability : 
 VII. 1, Biot, account of the ‘ Mécanique Céleste.’ , 
 
 The last three volumes contain the debates, or conferences, be- 
 tween the teachers and their pupils, of which there are three in the 
 first of them, on arithmetic and algebra, not at all worth reading. 
 
 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 arithme- 
 tic. 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, 8, 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 he been kept at School, ever since they were capable of being 
 taught.” 
 
 Paris, eighteen-hundred. Condorcet. ‘Moyen d’ap- 
 prendre a compter surement et.avec facilité.’ Second edition. 
 Octavo (duodecimo size). 
 
 One of the simplest explanations of the most elementary arith- 
 metic 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 Geometria practica 
 de la real Academia de San Fernando.’ Octavo. 
 
 A very clearly written and printed work. 
 
WORKS ON ARITHMETIC. 83 
 
 Buckingham, eighteen-two ; second edition, eighteen-eight. 
 John King. ‘An Essay intended to establish a new uni- 
 versal 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 Trea- 
 tise on Arithmetic, rational and practical. Two volumes. 
 Octavo in twos. 
 
 This treatise has alist of a thousand subscribers, and has amused 
 me very much. The old notions of the style of a book were, it 
 seems, not extinct in Ireland a hundred years after they had been 
 exploded in England. The author, who handles his subject ably, 
 puts philomath after his name, and is perhaps the last of those who 
 rejoiced in a title which, though self-conferred, its owners would 
 not have changed for F.R.S. It is dedicated ‘to all those who 
 think that a knowledge of accounts is useful to mankind, from the 
 king on the throne to the lowest subject.’’ It has the praises of the 
 author’s friends in prose and poetry, duly prefixed. I quote a few 
 lines from one of the poets, desiring the reader to observe where 
 sophs come from, unsought, and whence Irish authors got their 
 stationery. 
 
 ‘¢ How many sophs, to sense and science blind, 
 Range through the realms of nonsense unconfin’d, 
 Unaw’d by shame, and unrestrain’d by law, 
 Their labour chaff, and their reward a straw; 
 Neglected and despis’d, they sink in shame 
 To that oblivion whence, unsought, they came. 
 
 The muse, indignant, oft with grief has seen 
 An author led by ignorance and spleen, 
 With snail-paced speed, but unremitting toil, 
 In attic chamber waste the midnight oil, 
 With waste of paper, loss of ink combined, 
 And pens from public offices purloined.*— 
 But DrereHAn of a more enlightened mind, 
 More innate genius, talents more refined,” &c. 
 
 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. 
 
 * Really I am afraid that there must have been some truth in this. Mr. Thomas 
 Moore gives a translation of the Pennis non homini datis of Horace, which shews that 
 he had heard of the thing at least. Of course he can clear himself: at any rate Lalla 
 Rookh has not much the air of having been written with a pen from a public office. 
 
84 A CHRONOLOGICAL LIST OF 
 
 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 Schwan- 
 pan, for a description of which see Peacock, p. 408. 
 
 Paris, eighteen-seven. J. B. V. ‘L’Arithmétique en- 
 seignée 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 con- 
 structed a sap. The lady has the awkward name of Madame Epi- 
 nogy ; 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 émivoia. Nevertheless the dialogues 
 are exceedingly good. 
 
 Paris, eighteen-eight. F. Peyrard. ‘Oeuvres d’Archi- 
 méde, traduites littéralement, avec un commentaire. Second 
 edition, two volumes. Octavo. 
 
 I mention this work, not only for the drenarius already noticed, 
 but for the disquisition by Delambre on the arithmetic of the Greeks, 
 which afterwards appeared in the ‘ Histoire de l’Astronomie An- 
 cienne,’ Paris, eighteen-seventeen, guarto, two volumes. Delambre 
 was a real reader of the works he cites. He collected his materials 
 from Nicomachus, the Theologumena, Barlaam, the two Theons, 
 Ptolemy, Eutocius, Pappus, and Archimedes. I may as well say 
 here what I have to say on those of the above who have not been 
 mentioned elsewhere. 
 
 The best edition of Ptolemy is that of Halma, which is a collec- 
 tion of Ptolemy and his commeutators, published at different times, 
 and separately, —the whole making distinct works, as well as a set. 
 Of this I know only four volumes, of which the two to the present 
 purpose are Paris, eighteen-thirteen and sixteen. ‘KAavduov Irode- 
 patov paOnparikn Svvragkis . . (Gr. Fr.) quarto, two volumes. 
 Brunet says that the Pomtiantewans of Theon, and the Kavoves 
 IIpoxerpot of Ptolemy, are in five more volumes, Paris, eighteen- 
 twenty-one, twenty-two, twenty-two, Dubay ict twenty-five, 
 quarto. 
 
 The commentaries of Eutocius on the works of Archimedes are 
 
WORKS ON ARITHMETIC, 85 
 
 to be found in several editions, but best in that of Joseph Torelli, 
 Oxford, seventeen-ninety-two, ‘’Apyyundous ra ow lomeva peta Tov 
 Evrokiov “Ackadovitou tropynpatov (Gr. Lat.). Folio. 
 
 The fragment of the second book of Pappus (the only part of 
 the first two books published as yet, if, indeed, any more exist) is 
 to be found (Gr. Lat.) in the third volume of John Wallis’s Opera 
 Mathematica, Oxford, sixteen-ninety-nine, folio in twos. 
 
 London, eighteen-ten. W. Tate. ‘A System of Com- 
 mercial 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 seven- 
 teen-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 with- 
 out a Knowledge of the Rule of Three ; and followed by Stric- 
 tures 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 pro- 
 perly arranged. 5. Or in which the rationale of arithmetical opera- 
 tions 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. 
 
 I 
 
86 A CHRONOLOGICAL LIST OF 
 
 Dublin, eighteen-twelve. John Walker. ‘The Philo- 
 sophy 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 Arith- 
 metical 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 Cole. 
 brooke. ‘Algebra, with Arithmetic and Mensuration, from 
 the Sanscrit of Brahmegupta and Bhascara.’ Quarto. 
 
 The first work has notes by S. Davis, and is from a Persian ver- 
 sion of Bhascara’s Sanscrit. The second work is also from the Per- 
 sian. The third, which contains not only the two works of Bhas- 
 cara, but also an arithmetical chapter from Brahmegupta, is all from 
 the Sanscrit. It is also pretty copious in selections from the Com- 
 mentators, and has a large body of dissertation by Colebrooke himself. 
 But it does not entirely supersede the former two, which have like- 
 wise valuable annotations. 
 
 Edinburgh, eighteen-thirteen. Elias Johnston. ‘A sure 
 and easy Method of learning to Calculate.’ Duodecimo in sizes. 
 
 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 com- 
 mercial purposes. At the end are some forms for letters of cere- 
 mony ard 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. 
 
WORKS ON ARITHMETIC. 87 
 
 London, eighteen-fourteen. S. F. Lacroix. ‘ Traité 
 Elémentaire d’Arithmétique a Pusage de l’école centrale des 
 quatre-nations.’ Tenth edition. Octavo. 
 
 A well-known work, by one of the most systematic and most 
 widely circulated of elementary writers. The sixteenth edition was 
 Paris, eighteen-twenty-three, octavo. There was an English trans- 
 lation, London, eighteen-twenty-three, octavo, anonymous, —an at- 
 tempt to introduce demonstrative arithmetic into our schools. 
 
 The third edition of an American translation, by John Farrar, 
 appeared Cambridge (U.S.), eighteen-twenty-five, octavo in twos. 
 
 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 sub- 
 jects, which are very good, and, as parts of a learner’s course, un- 
 exampled. 
 
 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, Ni- 
 comachus, Jamblichus, and Boetius. ‘Together with some re- 
 markable particulars respecting perfect, amicable, and other 
 numbers, which are not to be found in the writings of any 
 
88 A CHRONOLOGICAL LIST OF 
 
 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 anti- 
 cipation 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 frac- 
 tions are given. But, in the main, it is an application of the same 
 principle as that which has long been used in astronomical loga- 
 rithms, 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 130/.; of pounds up to 7 ewt. 16lbs.; of twelfths up to 333} ; 
 of sixteenths up to 3124; 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 26001. 
 
 Leipsic, eighteen-seventeen. Fred. Astius. ‘ Theolo- 
 gumena Arithmetice ....Accedit Nicomachi Geraseni In- 
 stitutio Arithmetica.’ Octavo. (See p. 17.) 
 
 These @coAoyotpeva 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 sizes. | 
 
WORKS ON ARITHMETIC, 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 com- 
 prehensive 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 Nu- 
 merical 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. ‘ Hore Arithmetice.’ 
 
 The first-mentioned paper contains the most remarkable addition 
 made to arithmetic in modern times, the value of which is gradually 
 becoming known. On this subject I may refer to Mr. Horner’s 
 paper on Algebraic Transformation in the Mathematician (vols. i. 
 and ii, various numbers); to J. R. Young, ‘An elementary trea- 
 tise on Algebra,’ London, eighteen-twenty-six, octavo, as the first 
 elementary writer who saw the value of Horner’s method; J. R. 
 Young, ‘ Theory and solution of Algebraical Equations of the 
 higher orders,’ London, eighteen-forty-three, octavo ; Thos. Stephens 
 Davies, ‘A Course of Mathematics .... by Charles Hutton,’ twelfth 
 edition, London, eighteen-forty-three, octavo ; 'T. S. Davies, ‘Solu- 
 tions of the principal questions in Dr. Hutton’s Course of Mathe- 
 matics,’ London, eighteen-forty, octavo ; Peter Gray, four papers in 
 the Mechanic’s Magazine for March, eighteen-forty-four; A. De 
 Morgan, ‘ Notices of the progress of the problem of Evolution,’ in 
 the Companion to the Almanac for eighteen-thirty-nine, with the two 
 articles headed ‘ Involution and Evolution’ in the Penny Cyclo- 
 peedia, and in the Supplement (eighteen-thirty-eight and forty-five), 
 and a Letter to the Editor of the Mechanic’s Magazine, published 
 in that work for February, eighteen-forty-six. 
 
 In connexion with this subject I ought to mention Mr. Thomas 
 Weddle’s ‘ New simple and general method of solving numerical 
 equations of all orders,’ London, eighteen-forty-two, quarto. This 
 is an organised process, in which the principle of each step is the 
 correction of the preceding result by multiplication, not by addition. 
 
 Edinburgh, eighteen-twenty. John Leslie. ‘The Phi- 
 losophy of Arithmetic.’ Octavo. | 
 
 I have spoken of this work in the Introduction. P. 373, 405, 
 411, 477. 
 
 Vienna, eighteen-twenty-one. Geo. Fred. Vega. ‘Vor- 
 lesungen tiber 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. 
 12 
 
90 A CHRONOLOGICAL LIST OF 
 
 Leeds, eighteen-twenty-three. [Walker ?] ‘Elements 
 of Arithmetic for the use of the Grammar School... .’ Duo- 
 decimo in threes. . 
 
 An excellent little work, which I suppose I am right in attri- 
 
 buting 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 Arith- 
 metic 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 Cyclopedia. 
 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. ‘Essen- 
 tials 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 Ele- 
 mentary 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. ‘ Illus- 
 trations of Theoretical Arithmetic.’ Duodecimo in sizes. 
 
 A demonstrated system of Arithmetic, at a time when there were 
 few such things in English. 
 
 London, eighteen-twenty-seven. HI. and J. Grey. ‘ Prac- 
 tical 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. 
 
 A clear and excellent work, written by a man of real science. I 
 
WORKS ON ARITHMETIC. 9] 
 
 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. ‘Flémens 
 d’Arithmétique.’ Sixth edition. Octavo. 
 
 More complete than Lacroix in details. Bourdon was an excel- 
 lent 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 im- 
 proved by the Rey. 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 Arithme- 
 tic, 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.’ 
 
 This is from the Encyclopedia Metropolitana. As part of the 
 first volume of the Pure Sciences, the date is eighteen-twenty-nine : 
 but it was separately published, in the parts, in eighteen-twenty-five 
 or twenty-six. This is the article mentioned in the Introduction. 
 I subjoin a few remarks, either in correction of some slips of the 
 pen, or in addition to what has been said. The paging is that of 
 the Encyclopedia Metropolitana. 
 
 P. 402, note. A sexagesimal table is sometimes pasted into a 
 work to which it does not belong, by some old owner. 
 
 P. 404. For ‘Regiomontanus, in his Opus Palatinum de Tri- 
 angulis,’ read ‘in his work de Triangulis ;’ and for ‘ as we learn from 
 the relation of Valentine Otho, zn his preface to that work,’ read ‘ in 
 his preface to the Opus Palatinum.’ But further observe, that though 
 Regiomontanus did change the usual sexagesimal radius into a de- 
 cimal one, no decimal tables of his were published until some time 
 after Apian had published decimal tables of his own. At least, after 
 much research, I can find none even mentioned. It is a mistake (a 
 
92 A CHRONOLOGICAL LIST OF 
 
 universal one) to say that decimal sines were published with the 
 work de Triangulis, in 1533. It arises from such tables being in the 
 second edition of that work, in 1561. 
 
 P,411. Add that Chaucer, in his work on the Astrolabe, makes 
 augrime figures to be exclusively the Arabic numerals. 
 
 P. 414. The work of Pacioli (not Paccioli, though very often so 
 spelt) was published in 1494 (not 1484). Canacci, not Caracci. 
 
 P. 419 (note). Besides the Fasciculus Temporum (of which, by 
 the way, there are editions dated 1474, and two or three without 
 date, probably earlier, according to Hain), there are the d/manac 
 and Ephemerides of Regiomontanus, described in the Companion 
 to the Almanac for 1846, with extensive masses of Arabic numerals. 
 
 P. 425. Note that there is an express commentary by Jam- 
 lichus on Nicomachus, and that the former probably did not write 
 the Theologumena at all. 
 
 P. 434. I have no doubt that Dr. Peacock has authority for 
 saying that the old English Arithmeticians called the then common 
 mode of dividing the scratch way (as indeed it was); but I have 
 never met with the phrase. 
 
 P. 438 (note). It was quite common, even before the invention 
 of printing, to speak of Algus as the inventor of decimal notation, or 
 Algorithm : and several of the ornamented title-pages of Simon de 
 Colines, the successor of Henry Stephens, have figures of Ptolemy, 
 Orpheus, Euclid, and dlgus, on one side, opposite to the Muses of 
 Astronomy, Music, Geometry, and Arithmetic, on the other: as if 
 paired for a dance. 
 
 P.440. See what I have said on the Disme of Stevinus (1585) ; 
 and note that Simon of Bruges is Stevinus himself. 
 
 P. 442. There is nothing about decimal fractions in Hunt’s 
 Handmaid. 
 
 P. 444. I suggest as the derivation of furlong, a corruption of 
 forty-long. The ordinary derivation, furrow-long, can hardly have 
 a foundation in fact; for the length of a furrow depends upon that 
 of a field. 
 
 P. 454. For ‘ Wingrave’ read ‘ Wingate.’ As to the note, I 
 differ greatly (as appears elsewhere) from Dr. Peacock’s opinion of 
 Cocker. With respect to the last sentence : ‘It may be worth while 
 observing that this modest and useful book is not honoured with 
 poetical recommendations ;’ it would seem that the copy consulted 
 wanted the frontispiece, on which, under the portrait of Cocker, are 
 
 these lines : 
 “‘ Ingenious CockER ! (now to Rest thou’rt Gone) 
 Noe Art can Show thee fully but thine own, 
 Thy rare Arithmetick alone can show 
 Th’ vast Swms of Thanks wee for thy Laboure owe.” 
 
 P. 464. For ‘James Peele’ read ‘John Mellis,’ and alter the 
 date to 1588, as above. See on Stevinus what I have said above. 
 
 P. 471. The leuca was originally a measure of the Gauls; and 
 it may be much doubted whether the measure was ever out of 
 France since the time of Czsar. See the article League in the 
 Penny Cyclopedia. Surely Dr. Peacock must have written ‘France’ 
 
WORKS ON ARITHMETIC. 93 
 
 meaning to have written ‘England.’ This alteration of one word 
 makes every thing right. 
 
 The small number of inaccuracies noted above are all I have 
 been able to lay my hands on, in going through Dr. Peacock’s most 
 valuable article, with the materials for this list about me. I may add, 
 that in two, or perhaps three instances, I remember to have found 
 an edition mentioned as the earliest which is not so. But to this 
 every writer is subject who has the courage to attempt history upon 
 such bibliographical guides to the sources as now exist. 
 
 London, [eighteen-twenty-nine|. James Parker. ‘ Arith- 
 metic and Algebra.’ Octavo. 
 
 An early work of the Society for the Diffusion of Useful Know- 
 ledge. 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 ques- 
 tions set, for instance, in multiplication have digits which descend 
 by twos, as in 86420, 20864, 75319, &c. So that it is gravely pro- 
 posed thatthe 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. ‘ Arith- 
 metic.’ Quarto. 
 
 This is the article so headed in Brewster’s Cyclopzdia, 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. Eng- 
 lish and Persian. 
 
 This algebra is very arithmetical. It was published by the 
 Oriental Translation Fund. M. Libri, in the work presently men- 
 tioned (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. 
 
94 A CHRONOLOGICAL LIST OF 
 
 Springfield (U. S.), eighteen-thirty-three. Zerah Col- 
 burn. ‘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 arith- 
 meticians 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 Devon- 
 shire 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 ration- 
 ale’... . Duodecimo in threes. Second edition. 
 
 The pretensions of this work are manifestly exaggerated ; but the 
 methods are ingenious. Books of mental arithmetic choose their 
 own examples, and thus make their own rules work well. The 
 guadrantal method (why so called I cannot guess) is making use 
 of a rule for turning pounds, &c. into farthings, which requires the 
 learning of a table ; and then making such applications as the follow- 
 ing :—A pound troy contains six times as many grains as there are 
 farthings in the pound; therefore the grains in 4 pounds troy are 
 the farthings in six times as many pounds sterling. Some of these 
 rules would really become very effective, if the method of deci- 
 malising the parts of a pound were used. Multiply the pounds by 
 a thousand, and subtract 4 per cent, and the result is the number of 
 farthings. Thus: 
 
 £1763 . 17. 92 is 1763°89062 
 
 176389062 
 Apercent  70555°62 
 
 Subtract 1693335 = No, of farthingsin £1763 . 17 . 93 
 
WORKS ON ARITHMETIC. 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’a la fin du 
 xvi’ siecle.’ 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 un- 
 finished; 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 In- 
 dorum dictus est (vol. i. pp. 804-376)—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 Geometrie of the same 
 author, being the whole of his algebra, written in twelve-hundred- 
 and-twenty (vol. ii. pp. 805-476) — 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 Com- 
 plete 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 arith- 
 metic 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 coun- 
 tries, 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 
 
96 A CHRONOLOGICAL LIST OF 
 
 method of solving equations. In eighteen-forty-one the same author 
 published an edition of Hutton’s Course itself. 
 
 London, eighteen-forty-two. ‘ Encyclopeedia 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 Tutor’s Assistant, or Comic Figures of 
 Arithmetic.’ Duodecimo in sizes. 
 
 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 phre- 
 nologised head. The worst of it is, that the joke will remain on hand 
 too long for the learner: a picture ofa 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. 
 
 Liege, eighteen-forty-four. EX. Forir. ‘ Issai d’un cours 
 de Mathématiques a Pusage des éleves du collége communal 
 deisIaéges.o! asi: 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.’ Duwodecimo. 
 
 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 Ele- 
 ments of Arithmetic.’ Fifth edition, with Appendixes. Duo- 
 decimo in threes. 
 
 The previous editions were eighteen-thirty, thirty-two, and thirty- 
 five, duodecimo, and eighteen-forty an 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. 
 
WORKS ON ARITHMETIC. 
 
 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. 
 
 Date. 
 
 SEVENTEEN- 
 
 -thirty-four . 
 -fifty-four . . 
 -fifty-seven . 
 -sixty-one. . 
 -sixty-two. . 
 -sixty-six .. 
 -sixty-six .. 
 -seventy 
 
 -seventy-one. 
 -eighty-eight 
 -eighty-nine . 
 
 EIGHTEEN- 
 
 -four 
 
 -five 
 
 -eight . 
 -eleven 
 -eleven 
 -twelve 
 -sixteen . , 
 -seventeen 
 -eighteen . 
 -eighteen. 
 -twenty 
 -twenty 
 -twenty-two. 
 
 twenty-three 
 
 | 
 
 | 
 
 | 
 
 | 
 
 | 
 
 -twenty-three 
 
 | -twenty-four 
 
 | twenty-six FE 
 -twenty-six . 
 -twenty-seven 
 -twenty-eight 
 -twenty-eight 
 -twenty-eight 
 
 -twenty-eight 
 [circa] 
 
 -twenty-nine 
 
 Place. 
 
 London 
 EXODs 5 
 London . 
 London . 
 London .. 
 Sheffield 
 Birmingham . 
 Exeter . 
 London 
 Birmingham . 
 Edinburgh 
 
 Bingham . 
 Shrewsbury . 
 Barisgu-c 
 London 
 Birmingham . 
 Paris 
 Edinburgh 
 London 
 London 
 Dundee . 
 Glasgow 
 
 | Paris . 
 
 Bedford . 
 Canterbury 
 
 Lausanne. 
 Corky.) 
 London 
 London . 
 London . 
 London 
 London .. 
 [London]. . 
 
 London .. 
 
 Paris 
 
 Author. 
 
 ALEX. WRIGHT .. 
 
 ‘i | JOSEPH THORPE . 
 
 RaGADESB Yenc sare 
 JoHN DEAN. 
 Ricu, RAMSBOTTOM . 
 | Joun EApon 
 
 Wm.CrRUMPTON . . 
 GoelvER « .<pecmacmas 
 WM. ScoTtt . . «.« 
 
 Wn. TayLor .. 
 
 { 
 
 |Wnm. GoRDON . . « 
 
 W. BurTtERMAN . . 
 GrO. BAGLEXY .. . 
 Epm. DEGRANGE. . 
 Jou. Harris WIcKS 
 Je WICH ARDS) to) 3 6 
 J.CL.OUVRIER DELILE 
 Riek AVN oan su caeowks 
 JamMES MorRIson . 
 JoH. MATHESON . 
 JAMES NICOLSON. . 
 C. Morrison . 
 
 J7 BaJUVIGNY.. 
 W.H. WHITE . 
 
 H. Mar LeeEn and } 
 | W. SMEETH. - 
 
 Em. DEVELEY. . . 
 Jou. PENROSE. .. 
 ANTH. PEACOCK . . 
 EWrtoiae 
 Wo. PHILLIPS 
 
 S. P. REyNoLps 
 Ros. FRAITER. . . 
 R. W. Woopwarp . 
 
 ANONYMOUS .., 
 
 Hele Denivarn as 
 
 K 
 
 Leading Word of Title. 
 
 Bractions: 6. uc ghee 
 Treatise ... 
 Decimal . 
 
 Rule of Practice 
 Fractions anatomised . 
 Guides. a <P 
 Decimals and Mensuration . 
 School-Assistant .. . 
 Newa System’... vamrenn 
 Guide a) ches 
 Epstitutes: .10.. «<, eee 
 Dic OSG s+ eww « 
 Assistant, . «men ceeds 
 ‘Pratiques <a. 
 
 Merchant’s Companion 
 Practical . es 
 Appliquée au Commerce. . 
 Beautiesar.we was ches 
 Concise System . . 
 Theory and Practice . 
 
 Modern Accountant 
 Young Lady’s Guide . 
 Application au Commerce 
 Complete Course . . 
 
 Assistant . . 
 
 DiEmilew sie sauce. « 
 ‘Treatise. §: . Aig: 
 Mentaivae = 'se ty te 
 Examinations . . 
 
 New and concise System 
 Practical vcucat 
 
 Short System ops 
 Guidemt.) lees at eee ns 
 
 Intellectual (Pestalozzian) . 
 
 D’aprés Pestalozzi 
 
 Form. 
 
 4-4 (small). 
 6-+6 (2d ed.) 
 12. 
 
 6-L6. 
 12. 
 
A CHRONOLOGICAL LIST OF WORKS ON ARITHMETIC. 
 
 Date. 
 
 EIGHTEEN- 
 -twenty-nine 
 -twenty-nine 
 -twenty-nine 
 -twenty-nine 
 -thirty . 
 -thirty-one 
 -thirty-two 
 -thirty-four . 
 
 -thirty-four . 
 -thirty-four . 
 -thirty-six 
 -thirty-six . 
 -thirty-seven 
 -thirty-nine . 
 -forty . 
 -forty-three . 
 -forty-five - 
 Without date 
 
 es 
 
 ee 
 
 Place. 
 
 Paris 
 London 
 London 
 London 
 London 
 London . 
 Edinburgh 
 London 
 
 London . 
 London 
 | Edinburgh 
 
 London 
 Cambridge 
 
 London 
 London 
 London 
 ; London 
 | Dartmouth 
 
 | Boston (U. 8.) 
 
 . 
 
 Newcastle-on-Tyne 
 . Curist. Known. Soc. 
 
 . 
 
 _DanrIeEt DowLine . 
 
 Author. 
 
 . A. SAVARY . 
 
 ANONYMOUS 
 
 Gro. Hutton 
 
 . | James Perry . 
 . | W. Pursry .« 
 
 Ricu. CHAMBERS. 
 
 . | Ricu. Mostey 
 
 . | Jos. BARTRUM 
 
 J. FELTON . 
 
 MERCATOR . 
 W. Watson . 
 ADAM TAYLOR 
 
 | Jon. Davrpson 
 
 - | W.C. Hotson . 
 
 Henry AULT . 
 
 Wm. JARvIs 
 
 | THos TuHompson. 
 
 . 
 
 . | Roz. CUNNINGHAM . 
 ' 
 
 Leading Word of Title. 
 
 Arithmétique .. 
 Concise Arithmetician 
 Improved System... 
 Middle Stage and Key . 
 Practical ‘ 
 Theory and Practice . . 
 
 | Text-Book Pear ts Pera (Ie tek. 4 
 
 Introduction’. ". 9... eae 
 
 Tb Usiness DONK ss ine eames 
 | Taught by Questions. . . 
 ra Elements=.1 55. sens eee 
 : Guide so lt oP tees es ate eae 
 
 FATIFMINetiC! aes sats mer teeny 
 Improved Method .. . 
 
 PEE LOCI DIOS. ae 1 ae men nee 
 
 Expeditious Calculation. . 
 School (sus. t-0 toutes eee 
 [Useful Arithmetic] & Sequel 
 Requisites of Business . 
 
 . Catechism éMeaua.- is Rte 
 
 444, 
 
 6+6 (2d ed.) 
 444, 
 6-+6(4th ed.) 
 12, 
 
 6+6 (pt. 1). 
 8 (small). 
 
ADDITIONS. 
 
 —»~— 
 
 No place, date, nor author’s nor printer’s name. ‘ Algo- 
 rithmus 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 com- 
 pendiose sine figurarum (more Italorum) deletione compilatus. 
 artem numerandi omnemque viam calculandi enucleatim bre- 
 vissimé 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 quivis modica adhibita diligentia omnem cal- 
 culandi modum facillime adipisci potest.’ Quarto in fours. 
 
 Three different editions of this work are mentioned by biblio- 
 graphers, 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. 
 
 This is one of the editions alluded to in page 2, and for the 
 power of mentioning it from inspection (as well as the works of 
 Ghaligai and Texeda following) | am indebted to Dr. Peacock. It 
 begins with the following verses : 
 
 Chi de arte matematiche ha piacere 
 Che tengon di certeza el primo grado 
 Avanti che di quelle tenti el vado 
 Vogli la presente opera vedere 
 
 Per questa lui potra certo sapere 
 
 Se error sara nel calculo notado 
 
100 LIST OF WORKS ON ARITHMETIC. 
 
 Per questa esser potra certificado 
 
 A formar conti di tutto [sic] maniere 
 
 A merchadanti molta utilitade 
 
 Fara la presente opera e afatori 
 
 Dara in far conti gran facilitade 
 
 Per questa vederan tutti li errori 
 
 Ede iquaterni soi la veritade 
 
 Danari acquisterano e grandi honori 
 In la patria e de fuori 
 
 Sapran far le rason de tutte gente 
 
 Per le figure che son qui depente. 
 
 After a page of description of symbols, comes ‘Qui comenza la 
 nobel opera de arithmeticha ne laqual se tracta tute cosse amer- 
 cantia pertinente facta et compilata per Piero borgi da Veniesia.’ 
 At the end is ‘ Stampito in Veniexia per zovanne de Hallis 1488.’ 
 
 The author begins by saying that there are plenty of sufficiently 
 good masters, and not less abundance of most excellent authors. 
 His arrangement of the five acts of arithmetic is strange; they are 
 numeration, multiplication, partition, summation, and subtraction. 
 He uses the word million, and his numeration goes to millions of 
 millions of millions. He then proceeds to multiplication, and forms 
 a table of the nine multiples of 1, 2, ... 10, with those of 12, 16, 20, 
 24, 32, and 36. He then points out how to prove processes by 
 casting out sevens and nines; in doing which he shews how to 
 divide by 7, and uses the sum of the digits to find the remainder 
 to 9. Then follow multiplications done each in one line, even 
 when the multiplier has three or four figures. After this follows 
 the common method, in which the final process is addition, done 
 without any formal rule (in fact, addition is not made a formal rule 
 any where). Division is said, at the head of a chapter, to be done 
 in three ways; but only two are given: that per cholona, where the 
 whole is done in one line, and that per batelo, which is what I have 
 always designated by (O). Subtraction is then explained; and after 
 some applications, the rules for fractions (which are expressed in the 
 same way as now) are introduced. The rule of three (riegola del 
 tre) is given, and a large number of money applications. 
 
 Paris, fourteen-ninety-six. ‘In hoc opere contenta. Arith- 
 metica 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 men- 
 tioning 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 cus- 
 tomary to divide it in this manner,’so as to lead to the above imper- 
 fect description, The corrector of the press was David Lauxius, 
 and the printers John Higman and Wolfgang Hopilius. 
 
ADDITIONS. 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 Vine- 
 gia. 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 oppor- 
 tunity of seeing are the following, described by Hain and others : 
 
 Without date. ‘ Arithmetice Textus communis, qui pro Ma- 
 gisterio fere cunctis in gymnasiis ordinarie solet legi, correctus cor- 
 roboratusque perlucida quadam ac prius non habita commentatione 
 a Conrado Norico. Lipsiz per Martinum Herbipolensem.’ Fodio. 
 
 Without date, quarto, with the mark of Martin of Wurtzburg 
 (Herbipolis). ‘ Algorithmus linealis,’ beginning ‘ Ad evitandum mul- 
 tiplices 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 Arithmomachie,’ attributed to John (he should have 
 said William) Shirewood, bishop of Durham. Hain does not men- 
 tion this. 
 
 Babenberg, fourteen-eighty-three, dwodecimo. ‘ Rechnungsbiich- 
 lein.’ 
 
 Cologne, fourteen-eighty-five, quarto. ‘ Ars numerandi, seu com- 
 pendiosus tractatus de dictionibus numerabilibus.’ 
 
 Strasburg, fourteen-eighty-eight, quarto. Anianus. ‘ Compotus 
 manualis magistri aniani. metricus cum commento Et algorismus.’ 
 
 Leipzig, tourteen-eighty-nine, octavo. ‘ Rechnung auf alle Kauf- 
 mannschaft.’ Murhard and Kloss (who cites Panzer) give this bock 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, guarto. ‘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.) 
 
 K 2 
 
102 LIST OF WORKS ON ARITHMETIC. 
 
 Basle, fifteen-three. Wicolas de Orbelli. ‘ Cursus 
 librorum philosophie naturalis venerabilis magistri Nicolai de 
 Orbelli ordinis minorum secundum viam doctoris subtilis Scoti.’ 
 Quarto in fours. 
 
 This’ course of natural philosophy requires two pages of expla- 
 nation of arithmetical, and, in particular of Boethian, terms, and 
 nothing more. It purports to have been intended for those who 
 were studying theology. The Margarita Philosophica (page 5,) 
 seems to give the highest arithmetical limit in liberal education, 
 and the work before us the lowest. 
 
 See page 12. John de Lortse, &c. I was perfectly justified in 
 writing Lortse, for the name is so spelt in my copy in the clearest 
 black letter. Nevertheless, it is John de L’Ortie, and the book is a 
 French translation of the work of Juan de l’Ortega, mentioned by . 
 Dr. Peacock, pp. 426 and 436. Seeing de L’Ortie [sic] mentioned 
 by Meynier, and the word thus written being nothing but the 
 translation of L’Ortega (nettle), I compared the French work with 
 Dr. Peacock’s description of the Spanish, and found that the two 
 must necessarily be the same. On looking very narrowly at the f 
 which caused the mistake, I found in it a small. hook at the bottom, 
 which the letter f never has elsewhere in the work, and the letter i 
 alwayshas. The {then in Lortse is an 1 in which the dot (or rather 
 small line, for so it is in the black letter) has been accidentally 
 battered into the top of anf. But the alteration is so perfect that 
 every one reads it for an {. This is the second paragraph which 
 one battered letter has caused. In the new edition of Brunet it is 
 noted that Heber’s catalogue spells the word Lortse. My copy is 
 the one which was in Heber’s library. 
 
 This French translation gives Juan de ]’Ortega an earlier date 
 than the Spanish bibliographers, or those who have devoted them- 
 selves to the Dominicans, are aware of. Antonio, in his Spanish 
 lists, and Quetif, in his Dominican one, both give Ortega no earlier 
 date than 1534. But it appears that he was translated into French 
 in 1515. Panzer (xi. 465, according to Brunet) mentions a work 
 of L’Ortega, Messina, fifteen-twenty-two, beginning ‘ Sequitur la 
 quarta opera di arithmetica et geometria.’ 
 
 Florence (?), no date. Francisco di Lionardo Ghali- 
 gaio. ‘Summa De Arithmetica.’ Quarto in threes (N). 
 
 Florence, fifteen-sixty-two. Francesco di Lionardo 
 Ghaligai. ‘Pratica D’Arithmetica .... Nuovamente Ri- 
 vista.’ Quarto in fours. 
 
 Two editions of the same book. The first must have been pub- 
 lished before fifteen-twenty-three, the year in which Julius de Me- 
 dicis, to whom it is dedicated as Cardinal, would have been addressed 
 as Pope. It is a complete and advanced treatise of arithmetic, in 
 
ADDITIONS. 103 
 
 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. ‘<Li- 
 bretto di Abaco novamente Stampato: Composto per lo ex- 
 cellente maestro Joanne Francisco dal sole Ingegnero: utilis- 
 simo 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||777q°200||000, where || 
 (the two lines should be joined at the bottom) stands for mill, and 
 q° 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 com- 
 mercial. Rules for fractions are given, not complete. The rule of 
 three opens with an invocation of the Trinity : Sfortunati goes fur- 
 ther, 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 Her- 
 ford.’ 
 
 London, fifteen-seventy-four (printed by John Awdeley). 
 ‘ An introduc- [sic] 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) (Q). 
 
 These are, no doubt, reprints of the work of nearly the same 
 title, mentioned by Dr. Peacock (p. 419) as having been printed 
 at St. Albans in 1537. It was a good predecessor of Recorde’s 
 Grounde of Artes, but it makes only a poor follower. 
 
104 LIST OF WORKS ON ARITHMETIC. 
 
 See page 19. In speaking of Rudolph, I had overlooked that 
 Dr. Peacock (p. 424, note) describes his Algebra from inspection, 
 as having been edited by Stifel himself in 1571. 
 
 Rome, fifteen-eighty-three. Christ. Clavius. ‘ Epitome 
 Arithmeticee Practice.’ Octavo (small) (O). See page 33. 
 
 Witeberg, sixteen-four. Gemma Frisius. ‘ Arithmetice 
 Practicee 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 a la maniére d’Italie en 
 domaine et finance ordinaire : contenant ce en quoi s’exerce le trés 
 illustre et trés excellent Prince et Seigneur Maurice Prince d’Orange, 
 &c.’ 
 
 Frankfort, sixteen-thirteen. Joh. Faulhaber, ‘Ansa 
 inauditee et mirabilis nove 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.pc.vvs11., and it becomes 1613. 
 
 London, sixteen-fourteen. James Dowson. ‘De Nu- 
 merorum Figuratorum Resolutione.’ Octavo. 
 
 This work gives Pascal’s table. 
 
 No place nor date. Joh. Harpur. ‘The Jewell of Arith- 
 metick: or, the explanation of a new invented arithmetical 
 Table....* Quarto. 
 
 Perhaps this work is the Merchant's Jewel mentioned by Bridges 
 (p. 44) as of 1628, a date which, from the character of the work, I 
 should be disposed to give it. I can find no mention nor use of 
 decimal fractions in it. The operations are performed on a kind of 
 abacus. 
 
 Johnson, presently mentioned, describes a ‘ most excellent in- 
 strument invented by Mr. William Pratt, called, The Jewell of 
 Arithmatick ;’ which is perhaps the one described in the work 
 above, though with a different inventor’s name. 
 
 London, sixteen-thirty-three. Joh. Johnson (Survaighor). 
 ‘Johnson’s Arithmatick In 2 Bookes. The first, of vulgare 
 Arithma: with divers Briefe and Easye rules: to worke all the 
 
ADDITIONS. 108 
 
 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 (0). 
 
 In his decimal fractions, Johnson has the rudest form of nota- 
 
 tion ; for he generally writes the places of decimals over the figures, 
 thus— 
 1.2.3.4.5. 
 146°03817 would be 14603817 
 
 Otherwise, his system is tolerably complete. 
 
 London, sixteen-forty-nine. R.B. (Mr of Arts). ‘ Arith- 
 metick 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 Arith- 
 metick at first Designed for the use of the Free-Schoole at 
 Thurlow in Suffolk. By R. B. Schoolmaster there.’ Octavo (QO). 
 
 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 Arithme- 
 ticee ; the Numbering Nines. Or, Pythagoras His Table ex- 
 tended 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. 
 
106 LIST OF WORKS ON ARITHMETIC. 
 
 London, sixteen-eighty-five. John Wallis. ‘A treatise 
 of Algebra, both Historical and Practical.’ Folio in twos. 
 
 London, sixteen-ninety-three. John Wallis. ‘De Al- 
 gebra Tractatus ; Historicus et Practicus. Anno 1685 Anglice 
 editus ; Nunc Auctus Latine. folio in twos. 
 
 The Latin, or augmented edition, is the second volume of Wal- 
 lis’s collected works. See the Introduction to this Catalogue. 
 
 London, sixteen-ninety-seven - ‘Computatio Uni- 
 versalis, 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,*) 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 Biicherkentnis. Eilftes Stiick.’ 
 Octavo (small). 
 
 See the Introduction. This part contains the arithmetical biblio- 
 graphy; but the paging does not begin after that of the ‘ Zehntes 
 Stiick.’ In fact, the bibliography of this book of bibliography is a 
 study of itself. 
 
 Géttingen, seventeen-ninety-six, ninety-seven, ninety-nine, 
 eighteen-hundred. Abr. Gotthelf Kastner. ‘ Geschichte 
 der Mathematik... .’ Four volumes octavo. 
 
 See the Introduction, as to Kastner. 
 * T have seen this book sold with my name upon the cover as the author. To this 
 
 Ihave no objection, except my knowledge of the fact that it is the joint production 
 of Sir John Lubbock and Mr. Drinkwater-Bethune. 
 
LIST OF 1580 NAMES 
 
 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 asterisk prefixed shews that 
 the author has been referred to by Dr. Peacock in his History of Arithmetic. 
 
 JerlaeM,. At 68. 
 Abercromby, 51. 
 
 Abraham, Aben Ezra, 95. 
 
 Abraham, Chaja. 
 Abraham of Prague. 
 Abt. 
 Adami. 
 Adams. 
 Addington. 
 
 * Adelbold. 
 Aenez. 
 Agrippa, 13, 21. 
 Ahlwardt. 
 Albert. 
 Alberti. 
 Aldhelm. 
 
 Alexander de Villa Dei. 
 
 Alfraganus. 
 
 *Alous, 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’Autrépe. 
 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. 
 
108 LIST OF NAMES OF AUTHORS. 
 
 Barozzi. Bezout, 87. 
 Barreme, 75. *Bhascara Acharya, 86. 
 Barres, des. Bidder, 94. 
 Bartel. Biagio da Parma. 
 Bartjen, 48. Biermann, 
 Bartiens. Bierogel. 
 Barth. Bildner. 
 Bartl. Biler. 
 
 * Bartschius. Billy, De. 
 Bartoli. Billingsley. 
 Barton. Binet. 
 Bartrum, 98. Birkin, 80. 
 Barwasser. Birkner. 
 Basedow. Dirks Yo: 
 Baselli, 77. Bischof. 
 Bassi. Blagrave, 25. 
 Baum. Blasing. 
 Bausser. Blasius. 
 Bayer. Blassiére. 
 Bayley. Blavier, 75. 
 Beasley. Bluhme. 
 Beausardus. Blundevile, 9, 30. 
 Becker. Bobert. 
 Beckman. Bockel. 
 
 * Beda, 34. Bocmann. 
 Bedwell, 35. Bode. 
 Behen. Boden. 
 Behm. Boeclerus. 
 Behmen. *Boethius, xx. 3, 4, 5, 10, 11, 13, 
 Behrens. 17, 100,101. 
 Bellie. Bohm. 
 Benedictus, J. B., 31. Boissiére. 
 Benese, 18. * Bombelli. 
 Berckenkamp. *Bonacci (Leonard of Pisa). 
 Berger. Boninus. 
 Berghaus. Bonneau. 
 Bergstraste. Bonnycastle, 76, 91. 
 Berkeley, 63. *Borgi, Piero, 2, 98. ° 
 Bernard. Borgo, Pietro, 2. 
 Bernoulli, 78. *Borgo, Di (Paciolus), 2. 
 Berthevin, Boscherus. 
 Bertram. Boteler, 44. 
 Bettesworth, 71. Bouvelin. 
 Beutel. Bourdon, 91. 
 Beuther. Bouvelin. 
 
 * Beverege, 66. Bovillus, 3. 
 Bevern. Boyer. 
 Beyenberg. Boysen. 
 
 * Beyern. Bradwardine, 11. 
 Beyerus. *Bragadini, 11. 
 
LIST OF NAMES OF AUTHORS. 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. 
 Biichner. 
 *Buckley, vil. xvj. 20. 
 Budeeus, 20. 
 Bullialdus, 41. 
 *Bungus, 28. 
 Bunsen. 
 Bunzel. 
 Burckhardt. 
 Burtius. 
 Buscher, 28. 
 Bussche. 
 Busse. 
 Biitemeister. 
 *Buteo, 22. 
 Butler, 38, 87. 
 Butte. 
 Butterman, 97. 
 Buttner. 
 Buxerius. 
 
 Cadell. 
 Cesar (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. 
 
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. 
 *Cormmaro, 
 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. 
 
 LIST OF NAMES OF AUTHORS. 
 
 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. 
 
 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. 
 
 Davies, 19, 30, 89, 95. 
 
LIST OF NAMES OF AUTHORS. BBY 
 
 *Diophantus, 46. 
 
 Faber (Stapulensis), 3, 10. 
 Ditton, 66, 70. Falconius. 
 Dluskin, Famuel. 
 Dodson, 71, 73. Faulhaber, 104. 
 Dohren. Farrar, 87. 
 Donn, 72. Favre. 
 Doppler. Fayst. 
 Dowling, 74, 98. Feist. 
 Dowson, 104. *Feliciano da Lazesio. 
 Drape. Felicianus. 
 Dubost, 83. Felkel. 
 Duhren. Felton, 98. 
 Duke. Fenning, 71. 
 Duncombe, 78. Fermat, 47. 
 Dunkel. Fernel, 8, 15. 
 Dunus. Ferrel. 
 Dupp. *Fibonacci (Leonard of Pisa). 
 Diising. Finckius. 
 Dyer, 97. *Fineus (Orontius), vili. 5, 15, 16, 
 18, 19. 
 Fischer. 
 Fisher, 34. 
 Fissfeld. 
 Fletcher. 
 Eadon, 97. Flicker. 
 Eberenz. Flor. 
 Eggerer. Fludd, 35. 
 Ehrhardi. Fliigel. 
 Eichstadius. Follinus. 
 Elend. Fontaine. 
 Elf. Forbes, 48. 
 Elias Misrachi. Forcadel. 
 Emerson, 78. Forir, 96. 
 Enderlin. Fortia, 79. 
 Engelbert. Fortius, 16. 
 Engelbrecht. Forster, 38. 
 Engelken. Foster. 
 English. Foys de Vailois. 
 Eratosthenes. Fraiter, 97. 
 Erhorn. Francois. 
 Erxleben. Fraser, 80. 
 Eschenberg. Freber. 
 Eslingen, *Freigius. 
 *Euclid, 63, 77. Frend, 84. 
 Euler, 78. Frenicle, 53. 
 Euteneuer. Frey. 
 *Eutocius, 84. Friedeborn. 
 Everard, 68. Friesenborch. 
 Ewing. Friess. 
 
 *Frisius (Gemma), 25, 104. 
 
112 : LIST OF NAMES OF AUTHORS. 
 
 Frommichen. 
 Frommius. 
 Fuhrmann. 
 Fuller. 
 
 Funk. 
 Funcke. 
 Furit. 
 Furderer. 
 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, 
 Glaligas \ 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. 
 Gottsche. 
 Gottschling. 
 Gough, 79. 
 Graf. 
 Grafe. 
 Graff, De. 
 Grafte. 
 Graftinriedt. 
 Grammateus. 
 Grandi. 
 Graumann. 
 Grave. 
 Gravesande, S’, 73. 
 Gray, 89, 95. 
 Gregory, 85. 
 Greig. 
 Gretser. 
 Greve. 
 Grey, 90. 
 Griesser. 
 Grillet. 
 Grossi. 
 
 * Grostete. 
 
LIST OF NAMES OF AUTHORS. 
 
 Gruneberg. 
 Grunewald. 
 . Griining. 
 Griinmos. 
 Grupe. 
 Gruson. 
 Grysen. 
 Gryson, 
 Guden. 
 Gueinzius. 
 Gujieno (Siliceus), 15. 
 Gunter, xxv. 
 Gunther. 
 Giitle, 
 Guy, 
 
 Cy'He 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. 
 
 Hamel. 
 
 Hamelius (Paschasius), 
 
 Hamilton. 
 Hanel. 
 Hardcastle. 
 Hardy, 76. 
 Harpur, 104. 
 Harriot, 42. 
 Harris. 
 Harrison, 94. 
 Harroy. 
 Hartmann. 
 Harttrodt. 
 Hartwell, 22. 
 
 172 
 
 Hartwich. 
 
 Hase. 
 
 Hasius. 
 
 Hatton, 22, 66, 67. 
 Hauff. 
 
 Haughton, 38. 
 Haupt. 
 
 Hauser. 
 
 113 
 
 *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, 39, 
 Hennert. 
 Henning. 
 Henrici. 
 Hentsch. 
 Hentschel. 
 Hentz. 
 Hentzen. 
 Herberger. 
 Herbestus. 
 Herford, 103. 
 Herigone, 40. 
 Hermann. 
 Hermantes. 
 Herodian. 
 Herr. 
 Herrmann. 
 *Hervas. 
 Herwart. 
 Hesse. 
 
114 LIST OF NAMES OF AUTHORS, 
 
 Heyen. Hyperius. 
 Heyer. 
 
 Heynatz. 
 
 Hill, 70. 
 
 Hills (Hylles ?). 
 
 Hindenburg. 
 
 Hinrichsen. 
 
 Hirsch. Thne. 
 
 Hoar, 97. Illing. 
 
 Hobelius. Tmison. 
 Hochstetter. Imhof. 
 
 Hodder, 46, 57, 58, 59, 60. Imhoof. 
 
 Hodges. Impen. _ 
 Hodgkin, 67. *Ingpen, 36. 
 Hoff. Ingram, 85, 88. 
 Hofilin. Intelman. 
 Hoffmann. Ion. 
 
 Honenberg (Herwart). Irson. 
 Hohnstein. Isidorus. 
 Holiday. 
 
 Holland. 
 
 Hollenberg. 
 
 Holmes. 
 
 Holywood (Sacrobosco). 
 
 Holzmann (Xylander). 
 
 Holzwart. Jackson. 
 
 Hood, 24. *Jacob (Coburgk). 
 Hoorbeck. Jacobi. 
 
 Hope, 78. Jacobs. 
 
 Horem, 11. Jagar. 
 
 Horner, xxvii. 88, 89. Jager. 
 * Hostus. Jahn. 
 
 Hotson, 98. *Jamblichus, 17, 46, 88. 
 Howell. Jaquinot. 
 
 Howes. Jarvis, 98. 
 Hoyer. | Jaster. 
 
 Hiibsch. *Jeake, 55. 
 Hudalrich (Regius). Jesper. 
 
 Huet. *Joch. 
 
 Hugerus. *Johnson, 59, 104. 
 Hullie du Pont, L’. Johnston, 86. 
 Hume, 10. Joncourt. 
 Hunger. *Jones, 63, 69. 
 Hunichius. Jons. 
 
 Hunt, 39. Jordaine. 
 
 Hurry. Jordan. 
 
 Huswirt, 3. | Jordanus, 10, 100. 
 Huth. *Josephus Hispanus. 
 Hutton, 85, 89, 98. Josseaume. 
 
 Hylles, 31. MN Joyce, 90. 
 
LIST OF NAMES OF AUTHORS. 115 
 
 Jung. 
 Juvigny, 97. 
 
 Kaltenbrunner. 
 Kampke. 
 Kappell. 
 Karrer. 
 Karsten. 
 *Kastner, 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. 
 Kliigel. 
 Knoes. 
 Knorre. 
 Knott. 
 Kdbel, 9, 10, 11. 
 Koch. 
 Kochanskus. 
 Kohler. 
 Kolbe. 
 Konigsbrun. 
 Kénigstein. 
 Kopfter. 
 Koster. 
 
 Krafften. 
 Kraft. 
 Kramer. 
 Krause. 
 Kreitscheck. 
 Kretschmar. 
 Kritter. 
 Kroymann. 
 Kriiger. 
 Kriinitz. 
 Kruse. 
 Kundler. 
 Kupfer. 
 Kuster. 
 
 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. 
 Lehez. 
 
116 
 
 Leblond. 
 Lecchi. 
 Lechner. 
 Leeuwen, Van. 
 * Leibnitz. 
 Leichner. 
 Leiste. 
 Leistikov. 
 Lembke. 
 Lemoine. 
 Lempe. 
 Lenz. 
 
 *Leonard of Pisa (Bonacci Fibon- 
 
 acci), xviii. 1, 17, 94, 95. 
 Leotaud, 45, 
 Leri. 
 *Leslie, xvi. 89. 
 *Leupoldus. 
 Levera. 
 Levi. 
 Leybourn, 46, 49, 50, 52, 54, 
 © 072/89. 
 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, \ ao Stat 
 Lory. 
 
 LIST OF NAMES OF AUTHORS, 
 
 Losch. 
 Loscher. 
 Lossius. 
 Lostau, 67. 
 Lotter, 101. 
 Lovell. 
 Low. 
 
 Lowe, 70. 
 Lucar, 28. 
 Lucas. 
 
 Lucas a St. Edmondo. 
 
 Lucius. 
 Luders. 
 Ludolff. 
 Ludwig. 
 Lunenzius. 
 Lunesclos. 
 
 *Lunis, Gul, de, 102. 
 
 Lunze. 
 Luthardt. 
 Luya. 
 Lydal. 
 
 | , *Lyte, 36. 
 
 J SieweM., 62, 
 Machin, 74. 
 Maes. 
 Magelsen. 
 Mainwaring. 
 * Mair, 56, 77. 
 Malapertius. 
 
 *Malcolm, 66, 69. 
 
 Maler. 
 Malham. 
 Malleolus. 
 Mandey, 54. 
 Manning. 
 
 * Manoranjana. 
 Mantaureus. 
 Mapheeus. 
 Marcellus. 
 Marchetti. 
 Mariani. 
 Markham. 
 Marleen, 97. 
 
LIST OF NAMES OF AUTHORS. 
 
 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, 938. 
 
 Mole. 
 Molineux, 79. 
 Moller. 
 Monhemius. 
 Monson. 
 Monzon. 
 
 * Moore, 438, 49, 52, 59. 
 Morin. 
 Morisotus. 
 Morland, 47. 
 Morrison, 97, 
 Morsshemius. 
 Morssianus. 
 Moscopulus (Emanuel). 
 Mose. 
 Moser. 
 Moses. 
 Mosley, 98. 
 Miilich. 
 Miiller. 
 Mullerus. 
 Miillner. 
 Mullinghausen. 
 Munnoz. 
 Muris, J. de, 3, 4, 11. 
 
 Naiboda. 
 
 Nagel. | 
 
 Napier (Neper). 
 
 Nefe. 
 
 Nelkenbrucher. 
 
 Nemorarius (Jordanus). 
 *Neper, xxill. 35. 
 
 Nesius. 
 
 Neudorffer. 
 
 Neufville. 
 
 Nabod. \ 
 
 b17 
 
118 LIST OF NAMES OF AUTHORS. 
 
 Neumann. Overheyden. 
 Newton, 46, 73. Ozanam. 
 *Nicholas Smyrnezus Artabasda, 
 34. 
 Nicholson. 
 Nicolson, 97. 
 *Nicomachus (Gerasenus), 17, 
 46, 88. 
 Niese. WAP.. 30. 
 Nisbet. Pabst. 
 Nomer. Pachymeres, Geo. 
 Nonancourt. | *Paciolus (di Borgo), xx. 2, 95. 
 Nonius (Nunez), 23. *Pajottus. 
 Nottnagel. Palmquist. 
 Norfolk, 15. *Paolo dell’ Abacco. 
 Norry. *Pappus, 85. 
 *Norton, 27. Pardon, 68, 69. 
 Noviomagus, 18, 34. Parent. 
 Nunez, 23. Parker, 93. 
 Parisius. 
 Parrot. 
 Parsons, 63. 
 Partridge, 42, 51. 
 Pascal. 
 Obereit. Paschasius (Hamelius). 
 Ochler. Pasi. 
 Odo. Paterson. 
 Oelse. Patrick. 
 Oeser. Pauer. 
 Oesfeld. Paulinus. 
 Oesterle. Paulinus a St. Josepho. 
 Ofenlach. Paulus. 
 Ohm. | Paze. 
 Olaus. Peacock, xvil. xxiv. 14, 91, 97. 
 *Oldcastle, 28. *Peele, 28. 
 Opuntius. || Pelacanus. 
 Orbelli, 102. *Peletarius, \ 29. 95 
 *Orontius (Fineus), vill. 5, 15, Pelletier, fee 
 16, 18, 19. Pelicanus. 
 *Ortega, 102. Pell. 
 L’Ortie, J. de, 102. Penkethman, 41. 
 Oser. | Penrose, 97. 
 Ostermann. Da Perego. 
 Ostertag. Perez. 
 Osterwald. * Pergola. 
 Otto. Perry, 98. 
 *Oughtred, xiv. xxii. xxv. 37,|/ Pesare, Leon. de, 
 ob 40. Pescheck. 
 Ouvrard. Petersen. 
 Ouvrier de L’Isle. Petreius. 
 
LIST OF NAMES OF AUTHORS. 
 
 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. 
 Poschmann 
 Postellus. 
 Potter. 
 Pratt, 104. 
 Prestet, 54. 
 Preston, 88. 
 Prexendorffer. 
 Printz. 
 Pritchard, 90. 
 
 *Proclus. 
 
 Prosdocimo di Beldoman to. 
 
 Psellus, 21. 
 *Ptolemy, 84. 
 
 Purdon, 87. 
 
 Purser. 
 
 Putsey, 98. 
 
 Piitter. 
 *Pythagoras, 1, 4. 
 
 Quensen. 
 
 Rabus. 
 Rademann. 
 
 _ Raeymaker, 105. 
 { Ram. 
 
 | Ramsbottom, 97. 
 | *Ramus, 29. 
 Randall, 74. 
 Ranzovius. 
 Raphson, 66, 74. 
 Rawlin. 
 Rawlyns. 
 Raymaker. 
 
 Rea, 65. 
 
 Rees. 
 
 _Regius, Hudalrich, 16. 
 | Regneau. 
 Rehmann. 
 Reich. 
 Reiche. 
 Reichel. 
 Reimann. 
 Reimer. 
 Reinau. 
 Reiner. 
 | Reinhard. 
 | Reinhold. 
 Reinholdt. 
 Reisch, 4. 
 Reiser. 
 Remer. 
 
 119 
 
 Recorde, xxi. 21, 22, 59, 66. 
 
 *Regiomontanus, xiv. 16, 91. 
 
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. 
 Rover. 
 
 | 
 | 
 
 LIST OF NAMES OF AUTHORS. 
 
 Royer. 
 Rozensweig. 
 Ruchetta. 
 Ruddiman, 68. 
 *Rudolff, 23, 104. 
 Ruffus, 13. 
 Ruhl. 
 Rubmbaum. 
 Ruinellus. 
 Russell. 
 Rust. 
 Rutherford, 95. 
 Ryff, 9. 
 
 RaS.6., 62. 
 *Sacrobosco (Holywood, Hali- 
 fax), xix. 13. 
 Sadler, 75. 
 Salfeld. 
 *Salignac. 
 Salmasius. 
 Salomon, 75. 
 * Sanchez. 
 Sarret. 
 Sauer. 
 Saul. 
 *Saunderson, 68, 71. 
 Savary, 98. 
 Savonne. 
 Saxon. 
 Scalichius. 
 Schiffer. 
 Scharf. 
 Schatzberg. 
 Schedel. 
 Scheffelt. 
 Scheibel, xv. 106. 
 Scheilenberg. 
 Schelliesnig. 
 Schessler. 
 Scheubelius, 20. 
 Schey. 
 Schiener. 
 Schiller. 
 
 Schirmer. 
 
LIST OF NAMES OF AUTHORS. Lei 
 
 Schlee. 
 Schlegel. 
 Schleupner. 
 Schlonbach. 
 Schlosser. 
 Schligel. 
 Schliissel. 
 Schmalzried. 
 Schmid. 
 Schmidt. 
 Schmotther. 
 Schneidt. 
 Schonberger. 
 
 Schoner, 16, 29, 35. 
 
 Schoop. 
 Schottel. 
 *Schott, 45. 
 Schramm. 
 Schreckenberger. 
 Schreckenfuchsius. 
 Schreyber. 
 Schreyer. 
 Schroter. 
 Schiibler. 
 Schuler. 
 Schulze, 
 Schultze. 
 Schumacher. 
 Schtirmann. 
 Schurz. 
 Schwarzer. 
 Schweighauser. 
 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. 
 *Smyrnzus (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 a St. Gregorio. 
 Stephens. 
 Sterk, 16. 
 Stern. 
 *Stevinus, xxill, 26, 32, 35, 104. 
 Steyn. 
 Sthenius. 
 *Stifelius, 19, 23. 
 Stigelius. 
 Stillinger. 
 
 M 
 
122 LIST OF NAMES OF AUTHORS. 
 
 Stimmingen. 
 Stock. 
 Stoffler, 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. 
 Toissoniére. 
 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. 
 
LIST OF NAMES OF AUTHORS, 123 
 
 Vandenbussche, 23. 
 Vandendyck. 
 Vandervelde. 
 ¥*V asara. 
 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. 
 
124 LIST OF NAMES OF AUTHORS, 
 
 Wilson, 69. || Xenocrates. 
 Windssheim. Xylander, 21. 
 Wing, 53. 
 
 *Wingate, 42, 44, 48, 59, 73. 
 Winterfeld. 
 Wirth. Youle, 86. 
 Witekind. *Young, 89, 90, 93. 
 Witt, xxiv. 33. 
 Wittmer. 
 Wochenblatt. | 
 Wohlgemuth. Zaake. 
 Woit. Zabern. 
 Woldenberg. Zacharias. 
 Wolff, 68, 70. Zahn. 
 Wolphius. Zaragosa. 
 Woltmann. Zell. 
 Woodward, 97. Zeller. 
 Wordenmann. Zeplichal. 
 Worley. Zesen, Van. 
 Worsop. Zinge. 
 
 *Wright, 69, 97. ZLoéga. 
 Wucherer. Zon. 
 Wurffbain. Zonsen. 
 Wurster. Zubrod. 
 Wurstisius. Zucchetta. 
 Wybard, 43. | Zurner. 
 
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