VOL. LXVI.] PHILOSOPHICAL TRANSACTIONS. 731 



47 E. 1 , that the square of ac is equal to the square of ab multiplied by 2 ; the 

 square of ae equal to the square of ad or ac multiplied by 2 ; that is, equal to 

 the square of ab multiplied by 4, and so on. Thus the squares of ac, ae, ag, 

 Ai, &c. are respectively equal to the square of ab multiplied by the terms of the 

 following series 2, 4, 8, l6, 32, &c. where the 63d term gives the square of ab 

 multiplied by the last term of Sessa's Series for the Chessboard. 



If ex be drawn parallel to ap, the squares of Aa, a^, ac, a^, &c. will be res- 

 pectively equal to the square of ab multiplied by 3, 5, Q, 17, 33, 65, 129, &c. 

 Also if Ag be taken equal to \a, and ge be drawn parallel to bc, and this be re- 

 peated, the squares of Ae, &c. wiil be equal respectively to the square of ab 

 multiplied by 6, 12, 24,48, &c. And the squares on ao, &c. will be equal to 

 the square on ab multiplied by 4, 7, 13, 25, 49, &c. In like manner, if am be 

 taken equal to Ab, and mn be drawn parallel to bc, the squares on an, &c. will 

 be equal respectively to the square on ab multiplied by 10, 20, 40, 80, 160, &c. 

 And the square on as &c. will be equal respectively to the square on ab multi- 

 plied by the terms of the series, 6, 11, 21, 41, 81, 161, &c. 



In the same way, if right lines be drawn from e, e, g, n, i, &c. there will 

 arise numberless other series. And if bc be taken equal to ab multiplied by any 

 number, surd, fractional, or mixed, there will be obtained a great variety of 

 series, consisting respectively of terms, which are surd, fractional, or mixed. 

 And by dividing bc, de, ge, fg, mn, hi, &c. in different ways, according to plea- 

 sure, we may apply the same method to fractional numbers, without altering the 

 magnitude of bc. Thus, if bc be bisected, and a right line be drawn through 

 the point of bisection parallel to ap, there will be found right lines, the squares 

 on which are respectively equal to the square on ab multiplied by a great number 

 of fractions, having 4 for their comnlon denominator, and so on. 



Prop. 4. Tojind a right line, the square on which shall be equal to the square 

 on a given right line, divided by any number. — If, using the figure of the imme- 

 diately preceding problem, we suppose the given right line to be denoted by ai, 

 the squares on ah, af, ad, ab, &c. will respectively be equal to the square on ai 

 multiplied by ^, i, 4-, -iV, A> -rrs -i4-r. ttttj t4-5j toVt^ &c. or divided by 2, 4, 

 8, 16, 32, 64, 128, 256, 512, 1024, &c.; and so on for other numbers, whole, 

 surd, fractional, or mixed. 



Prop. 5. To cut off from a given right line a part expressed by any odd 

 number. — Let ab be the given right line, fig. 4. pi. 13. At right angles to it, 

 at one of its extremities b, draw an indefinite right line be. Let n be the given 

 odd number, expressing the part of ab to be cut off. Take bc such a right line 

 (prop. 3) that the square on it shall be equal to the square on ab, multiplied by 

 the number ■■ ~ . Draw cl as in the first theorem, and take ls equal to ab. 

 Then as is that part of ab, which is expressed by the otld number n. 



5 A 2 



