V1JI 



Till; P HOG K ESS OF 



brought by Gerberl into France. From that 

 (1 .to, then, we may reckon tlio introduction of 

 arithmetic, such as we have it at present, into 

 common use ; certainly one of the greatest boons 

 ever conferred on mankind. 



'Hie next great improvement introduced into 

 .irithmetic, was by John Muller, better known 

 ly his Latinized name of liegioraontauus. He 

 uas born in Konigsberg, a small town in Fran- 

 conia, in the year 1436. Scarcely had he reach- 

 ed his 14th year, when he became enamoured 

 with the charms of mathematics and astronomy. 

 He became the favourite pupil and friend of 

 1'urbach, who at that time was possessed of a 

 high reputation. With him he resided till the 

 time of his death, engaged chiefly in astronomi- 

 cal observations. The labours of Regiomonta- 

 nus were incessant, and the reputation which he 

 acquired was deservedly high. But we mention 

 his name here, because it was to him that 

 arithmetic owes the introduction of decimal 

 fractions. Thus he gave to numerical computa- 

 tion its utmost degree of simplicity and enlarge- 

 ment which it seems capable of reaching. 



Regiomontanus was incessantly occupied with 

 astronomical observations and calculations, which 

 of course, rendered the perfection of trigonome- 

 try an object of the utmost importance ; he did 

 accordingly bring it nearly to the state of sim- 

 plicity which it at present possesses. But the 

 calculations necessary for such purposes are 

 exceedingly laborious, and their labour neces- 

 sarily increased as astronomy advanced, because 

 it became essential to obtain more and more 

 accurate results. The sines and tangents of 

 angles could not be expressed with sufficient 

 correctness, without decimal fractions, extending 

 to five or six places ; and when to three such 

 numbers, a fourth proportional was to be found, 

 the work of multiplication and division became 

 exceedingly laborious. About the end of the 

 16th century, the time and labour necessarily 

 spent on such calculations, had become ex- 

 tremely burdensome to mathematicians and 

 astronomers. 



Napier of Merchiston, whose mind seems to 

 have had a bent towards arithmetical pursuits, 

 was the person to whom the happy thought oc- 

 curred of a method by which that labour might 

 be prodigiously diminished by substituting 

 addition and subtraction, for multiplication and 

 division ; this he did by the discovery of 

 logarithms, and the constructions of logarith- 

 mic tables by himself, Briggs, Gellibrand, &c. 



He observed that when the numbers to be 

 multiplied or divided, were parts of a geometri- 

 cal series, or progression, provided we know the 

 progression, the product or quotient might be 

 got at oace by inspection. Thus the 3d term 



of a progression, multiplied into the 5th term, 

 would make the 8th term ; and the 12th term, 

 divided by the 3d term, would make the 9th 

 term. He satisfied himself that all numbers 

 might be intercalated between the terms of a 

 geometrical progression ; and lie hit upon a 

 most ingenious way of proving the truth of his 

 proposition. What are called logarithms con- 

 sist of numbers in an arithmetical progression, 

 corresponding to all numbers supposed to exist 

 in a geometrical progression, and various sys- 

 tems of logarithms may be constructed. The 

 one which first occurred to the inventor, though 

 the simplest, was not so convenient, as the one 

 which occurred soon after to himself and his 

 friend Briggs, and according to which, the tables 

 now in use have been constructed. It is plain, 

 that if we add together two logarithms, the 

 logarithm constituting the sum of the two, will 

 correspond with the number which would be 

 obtained by multiplying the two numbers to- 

 gether , hence, by adding or subtracting lo- 

 garithms, and looking up for the new logarithm 

 in the tables, we find over against it, the num- 

 ber which would be obtained, if the numbers 

 whose logarithms we use, were multiplied or 

 divided by each other. What a prodigious 

 saving of time and trouble is thus accomplished, 

 roust be evident at first sight. Nor have the 

 benefits conferred upon mathematics by loga- 

 rithms been confined to this benefit, great as it 

 is ; they have spread themselves upon other 

 branches of the science. In 1618, Briggs pub- 

 lished the logarithms of the first 1000 numbers, 

 under the title of Logarithmorum chiliasprima ; 

 in 162-1, he published the logarithms of all 

 numbers, from 1 to 20,000, and from 90,000 to 

 100,000; all calculated to the 14th decimal 

 place. Death prevented Briggs from finishing 

 his plan, but it was completed by Gellibrand, 

 and published by him in his Trigonometria 

 Britannica, in 1633. 



II. GEOMETRY. 



We have seen the progress which geometry 

 had made among the ancients, and how Archi 

 modes by the method of exhaustions, had 

 succeeded in measuring spaces bounded by 

 curve surfaces. This method though satisfactory, 

 was exceedingly laborious, and being purely 

 synthetical, it did not enable us to apply it to 

 other discoveries of the same kind. A more 

 compendious, and more analytical method was 

 much to be wished for : and this great step was 

 made by Cavalleri, in the year 1635, in his 

 book entitled, Geometria indivisibilibus can. 

 tinuorwn nova quadam ralione promota. Ca- 

 valleri was born in Milan, in the year 1598. 



