May 4, 1893] 



NATURE 



17 



has been wisely made, yet it may happen that some important 

 particulars have become comparatively obscured under the new 

 treatment, that were in full light when the older plan was in 

 vogue. Since Harriot introduced into England the grand and 

 powerful improvement of making letters of the alphabet stand 

 for unknown quantities, school boys have been for the most part 

 regularly trained to look on algebra as a game of hide and 

 seek, where .r is concealed under conditions, and has to be 

 dragged out into the light. The idea of some undetermined 

 radix of a scale ofr notation, which was the very essence of the 

 algebra of Stifel and Stevin, has not been brought prominently 

 before them. It may be of interest to give four successive stages 

 by which a process of miiUiplication in algebra has arrived at 

 its present form. The first, originated by Stifel and adopted 

 by Recorde, made use of very strange signs with very odd 

 namei. In the product, beginning from the right, the first term 

 was called the absolute, the second the root, the third the square, 

 the fourth the oihe, the fifth the zenzizenzike, and the sixth the 

 sursolidc. In the second sta^e, Stevin's notation, adopted by 

 Briggs, is self-explanatory. The third system is Vieta's, adopted 

 by Harriot. 



- 2 



I 



^- 4 c? 



^ 



<? 



+ ^^ - 2 a€+ 3 ^ - 



4 ^ 



3 7f. 



- 4 



C? 



® - 2®+ 3® - 4 ' 

 © - - ' 



.0- 2®* 30-40 



*® - 2® * 3© - 4 



® -0.2rJ)-3®-0-4 



/p - 4 4^ 



aaa — 2aa 

 aa + a 



3« - 4 



aaaaa — laaaa -f Tjaaa - ^aa 



+ aaaa — 2aaa -f "^aa - ^a 

 aaa - 2aa + 3^ 



aaaaa - aaaa + 2aaa — ^aa 

 4 



a 



a" - 2a- r 3« 

 a^ + a + I 



ai - 



2a* + la^ 



- 4rt^ 



+ 32" - A" 



- 2rt- + 3ff - 4 



a* + 2d* 



3a- - 



a - 



As another example, a boy can use logarithms and understand 

 what they are, directly he has mastered the law of indices, but 

 in order to calculate them he imagines that he must know the 

 Binomial and Exponential Theorems. Surely it would aid him 

 to comprehend the relations of logarithms to numbers, if he 

 knew that they were originally calculated when the Binomial 

 and Exponential Theorems were 



given some slight sketch of the means by which they were then 

 determined. 



In the Daily News of December i6, 1892, a verse was quoted 

 as being often found written in a schoolboy's Euclid or 

 Algebra : — 



" If there should be another flood. 

 Hither for refuge fly, 

 Were the whole world to be submerged. 

 This book would still be dry." 



The schoolboy's charge of dryness must be met by showing 

 him how the progress of the arithmetic, geometry, algebra, 

 and trigonometry that he is learning has gone on in answer to 

 the needs that men have felt, and the desires they have formed. 

 There have been periods in which men, under the influence of 

 some widely-spread motive, have called for the aid of the 

 theorists to help them on their course, and the endeavour to 

 supply the great want of the time has brought about a great 

 advance in theoretical knowledge. As we look at the course of 

 these great movements, we find that it is the practical men that 

 supply the stimulu? to exertion, that set the few thinking for the 

 advantage of the many. Three instances of these great wants of 

 life — one of them row dead, the other two in ever- 

 increasing lifeand vigour,stand out prominently be- 

 yond the rest — astrology, commerce, and naviga- 

 tion. The induence of astrology exi ended over such 

 a vast period of time that we cannot trace its pro- 

 gress step by step from the ancient Chaldeans 

 to the Doctor Dee of the reign of Elizabeth, who 

 was the last eminent English mathematician of 

 the astrological sort, and at the same time one of 

 the great promoters of mathematics in its more 

 modern applications. We can see, however, 

 what has been left to us as the result of the 

 attention that was paid to astrology. The works 

 of Bhascara, himself an astrologer, show the 

 extent to which the Indian arithmetic and algebra 

 had gone, and what stock was in hand to be 

 turned to the new purpose of facilitating Euro- 

 pean commerce. We had also from these ancient 

 scholars the elements of trigonometry and tables 

 of sines and cosines. 

 The old astrologers were mainlalned and were enabled to 

 carry on their researches by the wealth of princes : Alphonso, 

 King of Castile; Frederick II., Emperor of Germany; 

 Matthias Corvinus, King of Hungary, are instances of monarchs 

 who had astrologers in their train, filling recognised positions 

 in their courts, Some of these were men of real learning ; 

 others, like Galeotti, introduced with the romance writer's 

 licence as to place and lime in Scott's " Quentin Durward," and 

 Lilly, who succesifully deluded the Parliamentary leaders in the 

 Civil War, were not much better than quacks. 



When we leave the astrological age and proceed to the com- 

 mercial, the history is much more complete and more interest- 

 ing. The whole story of the introduction of Indian arithmetic 

 into Europe by means of the Arabians, first as the result of the 

 Moorish conquests in .Spain, and then, after a long interval, as 

 a result of the commercial enterprise of Italy, is full of romantic 

 interest. It is curious to notice how strongly the commercial 

 element comes out in the algebra of Mahommed ben Musa. It 

 is all about questions of money, partnerships, and legacies. 

 When the practical objects for which mathematics were studied 

 became different, there was a corresponding alteration in the 

 mode liy which such researches were encouraged and main- 

 tained. There still remained the patronage of great princes 

 and nobles, but a new class of promoters arose among the greai 

 merchants and trading communities. A great wave of public 

 enthusiasm seems to have borne along with it all classes of 

 society, engaging them in the advancement of the new learning, 

 Benedetti held the office of mathematician to the Duke of Savoy, 

 with a good salary ; Torricelli was mathematician to the Duke 

 of Tuscany ; Harriot received £2^° ^ y^^f regularly from the 

 Earl of Northumberland, and while his noble patron was for 

 fifteen years in prison for complicity with some of the ambitious 

 plots of his friend Sir Walter Raleigh, Harriot, Hues, and 

 Warner bore him company, and were generally spoken of as 

 the Earl's three magi. As showing the interest taken by the 

 traders of great cities, it maybe noticed that some of the most 

 important treatises of the lime were written at the instigation ol 

 the merchants of Florence, and published at their expense. In 



NO. 1227, VOL. 48] 



unknown, and if he were our own country, the first English translation of Euclid was 



