1818.] A new Quadratic Theorem. 33 



placed within crotchets : and, thirdly, we multiply the last quo- 

 tient + 3 a x by 2 ; the product is + 6 a x, by which we divide 

 the third term -f 6 a • x 3 ; the quotient -f x 2 is placed under the 

 third term : fourthly, we square + x- = + x 4 , by which we 

 divide + x 4 ; nothing remains ; therefore the quantity is truly 

 measured by the square root 2 a° + 3 a x + x 2 . Mathema- 

 ticians must at once see the many advantages this simple theorem 

 has over the complex one at present in use, especially where the 

 higher geometry is connected with algebra. We remark that by 

 the old method we are obliged, as in the foregoing example, to 

 employ six lines, or 24 terms ; in the other only two lines, or 

 eight terms ; the saving in time and calculation is, therefore, 

 immense ; for in fact we change compound into simple division. 



Let us now work a few sums according to this method, be- 

 ginning with a trinomial. Extract the square root from 



a 4 — 4 a- x i + 4 at* 



a 2 — 2 x* + square root. 



Here the root of the first term a* is a 2 , which we place in the 

 quotient : secondly, we multiply this root by 2, giving for a 

 product + 2 a 2 , by which we divide the second term — 4 o 2 x 2 ; 

 the quotient is — 2 x% which we place under the second term : 

 thirdly, as this sum is a trinomial, we square the last quotient 



— 2 x 4 , giving + 4 x 4 , which, subtracted from 4 x 4 , leaves 

 nothing ; therefore the square root is a 2 — 2 x 2 -f- 0. 



Extract the square root from the following quadrinomial : 



a* - 4 a 3 b + 8 a b 3 + 4 b* 



a* — 2a b — 2 i 2 - square root. 



Here the root of the first term a* is a 2 , which we place in the 

 quotient: secondly, we multiply this root by 2, giving for a 

 product 2 a-, by which we divide the second term — 4 a 3 b; the 

 quotient is — 2 a b, which we place under the second term : 

 thirdly, — 2 a 6x2 = — 4 a b and + Sab 3 -. 4ab gives 



- 2 6 2 for the third term. Lastly, - 2 6 2 x — 2 b* = + 4 b*, 

 which subtracted from the last term leaves nothing. Therefore 

 the square root is a 2 — 2 a b — 2 6 4 — 0. 



This last example is taken from the Philosophy of Arithmetic, 

 an elementary work of considerable merit, by Mr. John Walker, 

 late Fellow of Trinity College, Dublin. 



Extract the square root of the following pentanomial : 



a* + 4 a 3 x I + 6 a 2 x« j + 4 a x 3 + x 4 

 a 4 +2ax| |+ x 2 +0 square root. 



Extract the square root from the following hexanomial : 



a« — 6 na +2za + 9nn\ — 6nz + zz 



a — 3 n I + z +0 square root. 



Vol. XI. N°I. C 



