NA TURE 



April g, 190J 



fashion against the existence of more than six principal 

 planets : — 



" Or si quelqu'un demande, pourquoy il n'y a que 

 six orbes des Pianettes, Keppler respond : — Parce qu'il 

 ne faut pas qu'il y ait plus de cinq proportions, tout 

 autant qu'il y a de corps reguliers is Mathematiques, 

 dont les costez et les angles sont esgaux les vns aux 

 autres. — Or six termes accomplissent le nombre de ces 

 proportions; et par consequent il n'y peut auoir que 

 six principales Pianettes." 



Could anything be more convincing? Perhaps, after 

 all, Uranus and Neptune are mere simulacra, will-o'- 

 the-wisps contrived by Satan to deceive a reprobate 

 race of astronomers no longer faithful to the great 

 principles of analogy. 



We have the authority of the Reverend Francois 

 Chevillard (1667) for believing that mathematicians 

 are (or should be) born under the sign of the Twins. 

 He says : — 



" Les Iumeaux. — Ce signe rend son homme beau, 

 misericordieux, sage, ingenu, libre, vn peu menteur, 

 coureur et voyageur, mediocre en commoditez, assez 

 fidelle pour estre Intendant des Finances, propre aux 

 Mathematiques, aux Loix, et a l'Arithmetique, sea- 

 chant dissimuler sa cholere, mais il sera pour courir 

 danger vers Page de trente-deux ans ou du feu, ou du 

 fer, ou de la morsure de quelque chien. ..." 



Here is something more properly mathematical. 

 John Abraham (1607) gives the product 6757x346 = 

 2337922, and after explaining the test by " casting out 

 the nines," proceeds as follows: — 



" Et d 'autant que la preuue de 9 n'est si certaine 

 que le contraire ou la preuue de 7 (sic). Nous auons 

 fait la preuue par 7. Et pour ce faire faut chasser les 

 7 dizaines de la somme a multiplier, sgauoir de 67 

 restent 4 de 45 restent 3 et de 37 restent 2 qu'il faut 

 poser a l'un des bras de la croix " (that is, the cross 

 used in the old-fashioned way of casting out the nines : 

 but Abraham's cross is like a big +), " puis en la forme 

 susdite faut aussi chasser les 7 du multiplieur, scauoir 

 de 34 restent 6 et de 66 restent 3 qu'il faut poser a 

 l'autre bras de la croix, et multiplier les deux figures 

 l'vne par l'autre, sgauoir 2 fois 3 sont 6 qu'il faut poser 

 sur le haut de la croix et pour la fin de la preuue faut 

 chasser les 7 des 2337922 de 23 restent 2 de 23 restent 

 encores 2 de 27 restent 6 de 69 restent 6 de 62 restent 

 6 et encores des 62 restent encores 6 qu'il faut poser au 

 bras de la croix. " 



It will be observed that this amounts to finding the 

 least positive residues of the factors with respect to the 

 modulus 7, and comparing their product with the re- 

 sidue of the product of the given numbers. The re- 

 sidues are found by actual division, not by any special 

 rule; curiously enough, it does not appear how the 

 author found the g-residues for the other test. No 

 proofs are given to justify the process in either case. 



The second part of Mr. Maupin's book (p. 160 to 

 end) deals mainly with the notes of Albert Girard to 

 the mathematical works of Stevinus. Both these men 

 were very competent mathematicians, and a study of 

 their work is very instructive. In their day, the science 

 of mathematics was but little advanced beyond the 

 stage at which it had been left by Pappus, Diophantus, 

 and Ptolemy; the notation of analysis was still very 

 imperfect; the methods of analytical geometry and in- 

 finitesimal calculus, as we now know them, had not 

 NO. 1745 VOL. 67] 



been invented; the prevailing style of demonstration, 

 as it appears to a modern reader, was both involved and 

 diffuse. But the times were ripening for the great 

 discoveries of Newton, Descartes, and Leibniz; and 

 if, as compared with the achievements of their imme- 

 diate successors, the work of men like Stevinus seems 

 poor and insignificant, we must remember that the 

 work of these humble pioneers was probably more im- 

 portant than appears at first sight. No one who has 

 studied the history of mathematics can have failed to 

 see how advance in the subject has accompanied im- 

 provement in notation. Now the essential features 

 of modern notation are due to the mathematicians of 

 the earlier part of the seventeenth century; and their 

 service in devising it is really considerable. Besides 

 this, they were the teachers of the younger mathemati- 

 cians of their time; and we may not unfairly credit 

 them with having done nothing to spoil and some- 

 thing to stimulate the minds of men with greater genius 

 than their own. 



The ingenuity of some of these old worthies, es- 

 pecially in diophantine analysis, is really remarkable, 

 and it is not always easy to see precisely their method 

 of procedure; for, after the manner of their time, they 

 publish results without demonstrations. Some very 

 curious results obtained by Girard (pp. 203-9 °f Mr. 

 Maupin's book) seem to show that he was acquainted 

 with the reduction of a quadratic surd to a periodic con- 

 tinued fraction; thus he obtains 103968 1/328776 as an 

 approximate value for V 10, and this rational fraction 

 is. in fact, the eighth convergent to the infinite con- 

 tinued fraction which represents V 10. G. B. M. 



ASTRONOMY FOR EXPLORERS. 

 Grundziige der astronomisch-geographischen Orts- 

 bestimmung auf Forschungsreisen. By Prof. Dr. 

 Paul Gussfeldt. Pp. xix4-368. (Braunschweig: 

 Vieweg und Sohn, 1903.) 



AS the field of the geographical explorer daily 

 narrows, so do the number and excellence of 

 books dealing with geographical exploration con- 

 tinually increase. The book under review treats of 

 the determination of time, latitude and azimuth with 

 a transit theodolite, and the methods described are the 

 simplest in use by the explorer; it will serve, however, 

 as an introduction to field astronomical methods 

 generally. 



The author leaves nothing unexplained, and com- 

 mences with elementary definitions of number and 

 quantity. A quarter of the book deals entirely with 

 elementary arithmetic, algebra, trigonometry and 

 analytical geometry. This is, perhaps, an excess of 

 thoroughness; for the explorer in most cases wants 

 to get to business as soon as possible, and if he has 

 ■not previously obtained a knowledge of the elements 

 of these matters, he is more than likely to be content 

 to use accepted formulae without investigation, so that 

 it is not quite clear for what class of reader the book 

 is written. 



It appears from the publishers' preface that Dr. 

 Gussfeldt has had considerable experience of field 



