2°'! S. X. Oct. G, 'dO.] 



NOTES AND QUEEIES. 



273 



printing him, use such symbols as sin 5° and tan^. 

 5° in the heading of tables. It is likely enough 

 that Euler's first use of the abbreviation, in for- 

 mvlce, will turn up in the middle of some one of 

 his earliest writings. But this is not of much 

 consequence : for the question is not who first 

 wrote sin A or tan A, instead of " sine of A" or 

 " tangent of A" ; but who first habitually intro- 

 duced ihQ functional symbol, be it "sin A" or 

 "sine of A,'"' into actual formula3,^ instead of taking 

 letters to represent the sines, cosines, &c. of angles 

 denoted by other letters ? And this is the real 

 meaning of the proposed question. 



In the Petersburg Transactions for 1729 (pub- 

 lished in 1735) Euler uses such symbols as/- AB, 

 cof: AB, cof (AB f AC). But only in descriptive 

 enunciation : in the actual problem he uses sepa- 

 rate letters. And G. W. Krafft, Euler's pupil, 

 does as much in the same volume. In the volume 

 for 1734-35 (published in 1740) the true invention 

 begins, sparingly used. "We see such a formula as 



(a+h) (a-b) 

 a — b cos.z 



In the volumes for 1736 and 1738 (1741 and 

 1747) there is moderate use of the invention by | 

 Euler, and some by Krafft. In the volumes for 

 1739, 1740, 1741-43 (1750, 1750, 1751) there is a 

 change in the character of the notation. Euler 

 now wants to distinguish between the sine to the 

 angle z and the angle to the sine z : this he 

 does by writing sin A . z and A . sin z ; and these 

 symbols are very frequently used. In the last 

 of the volumes there is also some return to the 

 simple abbreviations as now used. 



So far we have in succession the drops before 

 the shower, and the shower before the heavy rain. 

 And most of what I have given, though published 

 to the Academy of St. Petersburg, was not printed 

 in 1744, the year in which he printed the work 

 which first fully showed how great an improve- 

 ment had been proposed. 



In his llieoria Motuum Planetarum, Berlin, 

 1744, 4to. (the date is at the end), the whole 

 work is full of abbreviations for all the six trigo- 

 nometrical functions (sin, cos, tang, cot, sec, cosec), 

 with I prefixed for " logarithm," as in Z sin, I cos, 

 &c. The following symbols, 



^ g cos (»+<(>) a sin o < . sin iS i 



*—§•+§• cos iv-\-<py (cos ^vY' cos (;3 — a) i 



printed exactly as Euler gave them, will show 

 that our present notation would have given him 

 nothing to learn if he had died in 1744, and come 

 to life again in 1860. And we thus see that 

 Euler's proposal of our modern plan dates from 

 1734, and the constant and heavy use of it from 

 1744. In 1734, Simpson was writing his first 

 questions, in verse, for the Ladies' Diary, to say 

 nothing of election songs : this was before he 



came to London. None of his writings previous 

 to 1757 show any use of this notation. He pro- 

 bably first saw it in the writings of Clairaut, who, 

 being in England shortly after 1 750, paid him a 

 visit, and afterwards presented him with the book 

 on the Lunar theory. 



What Euler was is a question that cannot be 

 solved without calculation. His life, dating from 

 1736, the year in which his productions first began 

 to appear with rapidity, is a period of forty- seven 

 years : during the last seventeen of which he was 

 totally blind, and throughout the whole of which 

 he suffered from the consequences of a fever 

 which had deprived him of an eye. He was not 

 secluded from the world ; he married a second 

 wife, and was the father of thirteen children : 

 and this should stop the mouths of the biographers 

 who talk about Newton and others remaining 

 single that they might devote themselves to sci- 

 ence. Euler did more work than all of them put 

 together : so that any reflexions which are to be 

 cast upon matrimony must turn upon the quality 

 of the work, not upon its quantity. His life was 

 not exempt from those calamities which interrupt 

 the course of study. Ten children and twelve 

 grandchildren died before him ; his house was set 

 on fire and wholly burnt ; and an attempt to re- 

 store his sight by couching led to an illness which 

 nearly ended his days. He was fond of conver- 

 sation, of the society of his family, and of music : 

 and was, throughout the whole of his career, at- 

 tached to the court, and at the order, of a royal 

 or imperial patron. So little was there in his 

 manners of apparent unfitness for active life, that 

 in 1730, at twenty-three years old, when it seemed 

 likely that the Academy of St. Petersburg would 

 be dissolved, an admiral offered him a lieutenancy, 

 and promised him speedy promotion. Neverthe- 

 less, if his memoirs be counted, and if his separate 

 works (not volumes) be allowed for at the average 

 rate of twenty memoirs each, which is an insuffi- 

 cient rating both as to bulk and matter, the result 

 is as follows. Distribute Euler's work equally 

 through the whole period — which will be no great 

 alteration of the actual fact — and there is for each 

 and every fortnight in forty-seven years a separate 

 effort of mathematical invention, digested, arranged, 

 written in Latin, and amplified, often to a tedious 

 extent, by corollaries and scholia. Through all 

 this mass the power of the inventor is almost uni- 

 formly distributed, and apparently without effort. 

 There is nothing like this, except this, in the his- 

 tory of discovery : it is the thousand miles in the 

 thousand hours. 



There is a story among the traditions of his life, 

 whether in print or not I do not know, which ex- 

 plains how it was that several of his published 

 memoirs contain matter which memoirs published 

 before them had alreadycontalned in better and 

 more complete form. When he was at Berlin, 



