430 PHILOSOPHICAL TRAKSACTIONS. [aNNO 1753. 



space of 4 seconds ! On the whok, it may be concluded, that the several phe- 

 nomena, observed by this gentleman, in the transit and egress of Mercury, were 

 owing to indistinctness of vision, arising either from the eye, the telescope, or 

 the air; and that this alone may account for them all, without having recourse 

 to supposed circles of aberration ; which can never possibly exist in a well-con- 

 structed reflecting telescope. 



LVI. An Explanation of an Obscure Passage in Albert GirarcTs Commentary 

 on Simon Stevin's Works, p. 169, 170. By Mr. Simson, Prof. Math. Glas- 

 gow. Communicated by Philip, Earl Stanhope, p. 368. 



" Puis que je suis entre en la matiere des nombres rationaux, j'adjousteray 

 encore deux ou trois particularitez, non encor par cy devant practiquees, comme 

 d'expliquer les radicaux extremement pres, &c." 



The first thing Albert Girard gives in this place is a method of expressing the 

 ratio of the segments of a line cut in extreme and mean proportion, by rational 

 numbers, that converge to the true ratio. For this purpose he takes the pro- 

 gression 0, 1, 1, 2, 3, 5, 8, 13, 21, &c. every term of which is equal to the sum 

 of the two terms that precede it, and he says, any number in this progression 

 has to the following, the same ratio (nearly) that any other has to that which 

 follows it. Thus 5 has to 8 nearly the same ratio that 8 has to 13; conse- 

 quently, any 3 numbers next one another as 8, 13, 21, nearly express the seg- 

 ments of a line cut in extreme and mean proportion, and the whole line; so that 

 13, 21, 21, (n. B. 13 is wrong printed for the second number, instead of 21) 

 constitute near enough an isosceles triangle, having the angle of a pentagon; 

 i. e. whose angle at the vertex is subtended by the side of a pentagon in the circle 

 described about the triangle. 



Now this will be plain, if it be shown, that the squares of the numbers in 

 this series are alternately lesser and greater by an unit, than the product of the 

 two numbers on each side. Thus, in the 4 numbers, 5, 8, 13, 21, the square 

 of 8 is a unit less than the product of 5 and 1 3 ; but the square of 1 3 that next 

 follows 8, viz. 169, is a unit greater than 8 times 21, or 168; and so on con- 

 stantly. 



Case 1 . If a, b, c, be such numbers, that a -\- b = c, and ac = bb -\- 1. 



Then, if d be taken so that d = b -{- c; then shall bd-\- 1 = cc. For, be- 

 cause d =■ b + c; bd -{- 1 shall be ^ bb -\- be -\- 1 = ac -\- be, which is = 

 {a -\- b) X c = cc: ergo bd + 1 = cc. 



Case 2. If a, b, c, be such, that a -\- b= c, and ac -{- 1 = bh. 



Then, if d be taken so that d-= b -\- c; then shall bd=^ cc -{■ 1, For, be- 

 cause bd = bb -{- be z= ac -\- be -\- \ = {a + b) X c -j- 1 = cc -f 1. 



Problem. Having given the number a, in case 1 ; to find b and c, i. e. having 



