SvQ U 



divided into two equal parts, by a diagonal ; that the dia- 

 gonal of a fquare is incommenfiirable to the fide. 



Squares, for the Ratio of. They are to each other in 

 the duplicate ratio of their fides. E. gr. a fquare whofe 

 fide is double another, is quadruple of that other fquare. 



Square of the Cube. 7 p t> 



c yw p J r V J r ^ee rowEU. 



OQUARE of the Surdefolid. j 



Square Number, in Arithmetic and Algebra, is the pro- 

 duft arifing from multiplying any number by itfelf. Thus, 

 9 = 3.3, 16 = 4..4, 25 = 5. J, &c. are fquare numbers : 

 and fince, by the rules of algebra, — 3X — 3 = 95 — 4 

 X — 4=16; — JX — 5 = 25, &c. ; it follows, that 

 the fquare root of every pofitive number has two different 

 roots ; the one plus, or affirmative ; the other minus, or ne- 

 gative ; while the fquare root of every negative number is 

 imaginary, or impoflible. 



Square numbers have feveral remarkable properties, of 

 which the following are fome of the moil interefting, viz. 



I. Every fquare number is of one of the forms 4« or 

 4n -)- I ; that is, every fquare, when divided by 4, will 

 leave either o or i for a remainder : and underftanding this 

 expreflion ftill in the fame fenfe, the following table will 

 exprefs the forms of fquare numbers to the moduli, or 

 divifors, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.. 



2. And hence, by exclufion, we may derive the following 

 table of impoflible forms, viz. 



Thcfe formulae, as they involve no higher power of the 

 indeterminate n than the firft, arc called linear forms ; 

 but, by means of them, we eafily arrive at a variety of 

 quadratic formula:, which it is extremely ufeful to be ac- 

 quainted with, in pradifing the Diophantine or indeter- 

 minate analyfis. 



If a perfon, unacquainted with thefe exclufions, were 

 required to find two fuch numbers, that double the fquare 

 of the one, added to triple the fquare of the other, fhould be 



Vol. XXXIII. 



S Q U 



a fquare, he would fee nothing impoflible in the propofition, 

 and might, therefore, lofe many ufelefa hours in the re- 

 fearch ; whereas, by a little attention to the impoflible and 

 poflible forms, he would find the problem absolutely im- 

 poflible, and hence fpare himfelf much ufelefs labour. 



3. The method of deducing impoflible quadratic forms 

 from the linear ones above given, will be feeii immediately 

 from a fingle example. Let it be required to afcertaiu 

 whether the equation, 2 x' 4- 3 J"' = w ', be poflible or im- 

 poflible. Firlt, we may afl'ume », y, and lu, prime to each 

 other ; for if x and y have a common divifor, lu muit have 

 the fame, and the whole tquation may be divided by it ; 

 whereby it will be reduced to another equation 2 «" 4- 3 y"' 

 = tu' ', in which thefe quantities have no longer a common 

 divifor, or, in other words, they are prime to each other. 



Since, then, x and y are prime to each other, they can- 

 not be both of the form 3 n ; for, in this cafe, they would 

 have a common divifor 3. Let then, firft, x' be of the 

 form 3 n, and y' of the form 3 n 4- I ; then 2 x' is of the 

 *"orm 3 n', and 3 ji ' of the form 3 n' 4- 3, and confequently 

 their fum will have the form 3 n, which is impoflible, be- 

 caufe, in this cafe, to and x would have a common, divifor 3 j 

 and it we fuppofe x' of the form ^ n + 1, and y' of the 

 form 3 n ; then 2 .r ' 4- 3ji' is of the form 3 n 4- 2, which 

 is an impoflible form : and, lallly, if we an"ume both of the 

 form 3n + 1, then zx' 4- 3^' would have again the fame 

 impoflible form 3 n + 2 ; therefore, in no cafe can 2 .x' 4- 

 3jr- = to' be poflible in integral numbers. 



In the fame manner a variety of other impoflible forms 

 may be deduced, of which the following are thofe which 

 moll commonly occur. 



{5) Modulus 3. 



2/^ 4- 3 u' = w' 

 5/^ + 3a^ = w' 

 8/^ 4- 3B^ = to' 



2) /' 4- 31/^:^ TO- 



(6) Modulus 5. 



2/>' + 5 «' = TO^ 



3/' ± 5"' = '«'" 

 7 /^ + 5 u^ = TO^ 



2 ) t' ± 51/^ = TO ' 



(3^ + 



iSP ± 



Or the two latter general forms may be rendered more 

 comprehenfivc, by the introduftion of another indeter- 

 minate j; obferving only that, in this cafe, the quantity 

 mult always be prime to the modulus. With this condition, 

 the two latter may be written thus : 



(3;> + 

 Isp ± 



2) /^ 

 2) t^ 



4- 

 4- 



3 ja' = TO 



5 y u' = TO 



(7) Modulus 7. 



3) '' ± 7?"' = •" 

 5 '' ± 77"' = 

 6) r ± Tqu^ = 



(8) Modulus II. 



{lip + 2) /' ± II qu 



and 



7/ 

 [IP 

 [IP 



= TO- 



= to' 

 = to' 

 = to' 



(lip + 6) I' ± llqu' 

 (lip + 7) r ± llqu' 



(9) Modulus 13. 



(^iP ± 2) '' ± '3?"* = •»'* 



•3/ ± 5) '* ± '3?"* = *"* 



•3^ ± ^) '' ± '39"* = *"* 



'3/ ± 7) '* "t 'Sf * = '"* 



4"S 



(10) 



