The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 17

The line propounded beyng A.B. and deuided, as chaunce happeneth, in C. into ij. vnequall partes, I say that the square made of the hole line A.B, is equal to the two squares made of the same line with the twoo partes of itselfe, as with A.C, and with C.B, for the square D.E.F.G. is equal to the two other partial squares of D.H.K.G and H.E.F.K, but that the greater square is equall to the square of the whole line A.B, and the partiall squares equall to the squares of the second partes of the same line ioyned with the whole line, your eye may iudg without muche declaracion, so that I shall not neede to make more exposition therof, but that you may examine it, as you did in the laste Theoreme.

_The x.x.xvij. Theoreme._

If a right line be deuided by chaunce, as it maye happen, the square that is made of the whole line, and one of the partes of it which soeuer it be, shal be equall to that square that is made of the ij. partes ioyned togither, and to an other square made of that part, which was before ioyned with the whole line.

_Example._

[Ill.u.s.tration]

The line A.B. is deuided in C. into twoo partes, though not equally, of which two partes for an example I take the first, that is A.C, and of it I make one side of a square, as for example D.G. accomptinge those two lines to be equall, the other side of the square is D.E, whiche is equall to the whole line A.B.

Now may it appeare, to your eye, that the great square made of the whole line A.B, and of one of his partes that is A.C, (which is equall with D.G.) is equal to two partiall squares, whereof the one is made of the saide greatter portion A.C, in as muche as not only D.G, beynge one of his sides, but also D.H. beinge the other side, are eche of them equall to A.C. The second square is H.E.F.K, in which the one side H.E, is equal to C.B, being the lesser parte of the line, A.B, and E.F. is equall to A.C. which is the greater parte of the same line. So that those two squares D.H.K.G and H.E.F.K, bee bothe of them no more then the greate square D.E.F.G, accordinge to the wordes of the Theoreme afore saide.

_The x.x.xviij. Theoreme._

If a righte line be deuided by chaunce, into partes, the square that is made of that whole line, is equall to both the squares that ar made of eche parte of the line, and moreouer to two squares made of the one portion of the diuided line ioyned with the other in square.

[Ill.u.s.tration]

_Example._

Lette the diuided line bee A.B, and parted in C, into twoo partes: Nowe saithe the Theoreme, that the square of the whole lyne A.B, is as mouche iuste as the square of A.C, and the square of C.B, eche by it selfe, and more ouer by as muche twise, as A.C. and C.B. ioyned in one square will make. For as you se, the great square D.E.F.G, conteyneth in hym foure lesser squares, of whiche the first and the greatest is N.M.F.K, and is equall to the square of the lyne A.C. The second square is the lest of them all, that is D.H.L.N, and it is equall to the square of the line C.B. Then are there two other longe squares both of one bygnes, that is H.E.N.M. and L.N.G.K, eche of them both hauyng .ij. sides equall to A.C, the longer parte of the diuided line, and there other two sides equall to C.B, beeyng he shorter parte of the said line A.B.

So is that greatest square, beeyng made of the hole lyne A.B, equal to the ij. squares of eche of his partes seuerally, and more by as muche iust as .ij. longe squares, made of the longer portion of the diuided lyne ioyned in square with the shorter parte of the same diuided line, as the theoreme wold. And as here I haue put an example of a lyne diuided into .ij. partes, so the theoreme is true of all diuided lines, of what number so euer the partes be, foure, fyue, or syxe. etc.

This theoreme hath great vse, not only in geometrie, but also in arithmetike, as herafter I will declare in conuenient place.

_The .x.x.xix. theoreme._

If a right line be deuided into two equall partes, and one of these .ij. partes diuided agayn into two other partes, as happeneth the longe square that is made of the thyrd or later part of that diuided line, with the residue of the same line, and the square of the mydlemoste parte, are bothe togither equall to the square of halfe the firste line.

_Example._

[Ill.u.s.tration]

The line A.B. is diuided into ij. equal partes in C, and that parte C.B. is diuided agayne as hapneth in D. Wherfore saith the Theorem that the long square made of D.B. and A.D, with the square of C.D. (which is the mydle portion) shall bothe be equall to the square of half the lyne A.B, that is to saye, to the square of A.C, or els of C.D, which make all one. The long square F.G.N.O. whiche is the longe square that the theoreme speaketh of, is made of .ij. long squares, wherof the fyrst is F.G.M.K, and the seconde is K.N.O.M. The square of the myddle portion is L.M.O.P. and the square of the halfe of the fyrste lyne is E.K.Q.L. Nowe by the theoreme, that longe square F.G.M.O, with the iuste square L.M.O.P, muste bee equall to the greate square E.K.Q.L, whyche thynge bycause it seemeth somewhat difficult to vnderstande, althoughe I intende not here to make demonstrations of the Theoremes, bycause it is appoynted to be done in the newe edition of Euclide, yet I wyll shew you brefely how the equalitee of the partes doth stande. And fyrst I say, that where the comparyson of equalitee is made betweene the greate square (whiche is made of halfe the line A.B.) and two other, where of the fyrst is the longe square F.G.N.O, and the second is the full square L.M.O.P, which is one portion of the great square all redye, and so is that longe square K.N.M.O, beynge a parcell also of the longe square F.G.N.O, Wherfore as those two partes are common to bothe partes compared in equalitee, and therfore beynge bothe abated from eche parte, if the reste of bothe the other partes bee equall, than were those whole partes equall before: Nowe the reste of the great square, those two lesser squares beyng taken away, is that longe square E.N.P.Q, whyche is equall to the long square F.G.K.M, beyng the rest of the other parte. And that they two be equall, theyr sydes doo declare. For the longest lynes that is F.K and E.Q are equall, and so are the shorter lynes, F.G, and E.N, and so appereth the truthe of the Theoreme.

_The .xl. theoreme._

If a right line be diuided into .ij. euen partes, and an other right line annexed to one ende of that line, so that it make one righte line with the firste. The longe square that is made of this whole line so augmented, and the portion that is added, with the square of halfe the right line, shall be equall to the square of that line, whiche is compounded of halfe the firste line, and the parte newly added.

_Example._

[Ill.u.s.tration]

The fyrst lyne propounded is A.B, and it is diuided into ij.

equall partes in C, and an other ryght lyne, I meane B.D annexed to one ende of the fyrste lyne.

Nowe say I, that the long square A.D.M.K, is made of the whole lyne so augm?ted, that is A.D, and the porti annexed, y^t is D.M, for D.M is equall to B.D, wherfore y^t long square A.D.M.K, with the square of halfe the first line, that is E.G.H.L, is equall to the great square E.F.D.C. whiche square is made of the line C.D. that is to saie, of a line compounded of halfe the first line, beyng C.B, and the portion annexed, that is B.D. And it is easyly perceaued, if you consyder that the longe square A.C.L.K. (whiche onely is lefte out of the great square) hath another longe square equall to hym, and to supply his steede in the great square, and that is G.F.M.H. For their sydes be of lyke lines in length.

_The xli. Theoreme._

If a right line bee diuided by chaunce, the square of the same whole line, and the square of one of his partes are iuste equall to the lg square of the whole line, and the sayde parte twise taken, and more ouer to the square of the other parte of the sayd line.

_Example._

[Ill.u.s.tration]

A.B. is the line diuided in C. And D.E.F.G, is the square of the whole line, D.H.K.M. is the square of the lesser portion (whyche I take for an example) and therfore must bee twise reckened.

Nowe I saye that those ij. squares are equall to two longe squares of the whole line A.B, and his sayd portion A.C, and also to the square of the other portion of the sayd first line, whiche portion is C.B, and his square K.N.F.L. In this theoreme there is no difficultie, if you csyder that the litle square D.H.K.M. is .iiij. tymes reckened, that is to say, fyrst of all as a parte of the greatest square, whiche is D.E.F.G. Secondly he is rekned by him selfe. Thirdely he is accompted as parcell of the long square D.E.N.M, And fourthly he is taken as a part of the other long square D.H.L.G, so that in as muche as he is twise reckened in one part of the comparis of equalitee, and twise also in the second parte, there can rise none occasion of errour or doubtfulnes therby.

_The xlij. Theoreme._

If a right line be deuided as chance happeneth the iiij.

long squares, that may be made of that whole line and one of his partes with the square of the other part, shall be equall to the square that is made of the whole line and the saide first portion ioyned to him in lengthe as one whole line.

_Example._

[Ill.u.s.tration]

The firste line is A.B, and is deuided by C. into two vnequall partes as happeneth. The long square of yt, and his lesser portion A.C, is foure times drawen, the first is E.G.M.K, the seconde is K.M.Q.O, the third is H.K.R.S, and the fourthe is K.L.S.T. And where as it appeareth that one of the little squares (I meane K.L.P.O) is reckened twise, ones as parcell of the second long square and agayne as parte of the thirde long square, to auoide ambiguite, you may place one insteede of it, an other square of equalitee, with it. that is to saye, D.E.K.H, which was at no tyme accompting as parcell of any one of them, and then haue you iiij. long squares distinctly made of the whole line A.B, and his lesser portion A.C. And within them is there a greate full square P.Q.T.V. whiche is the iust square of B.C, beynge the greatter portion of the line A.B. And that those fiue squares doo make iuste as muche as the whole square of that longer line D.G, (whiche is as longe as A.B, and A.C. ioyned togither) it may be iudged easyly by the eye, sith that one greate square doth compreh?d in it all the other fiue squares, that is to say, foure long squares (as is before mencioned) and one full square. which is the intent of the Theoreme.

_The xliij. Theoreme._

If a right line be deuided into ij. equal partes first, and one of those parts again into other ij. parts, as chauce hapeneth, the square that is made of the last part of the line so diuided, and the square of the residue of that whole line, are double to the square of halfe that line, and to the square of the middle portion of the same line.

_Example._

[Ill.u.s.tration]

The line to be deuided is A.B, and is parted in C. into two equall partes, and then C.B, is deuided againe into two partes in D, so that the meaninge of the Theoreme, is that the square of D.B. which is the latter parte of the line, and the square of A.D, which is the residue of the whole line. Those two squares, I say, ar double to the square of one halfe of the line, and to the square of C.D, which is the middle portion of those thre diuisions. Which thing that you maye more easilye perceaue, I haue drawen foure squares, whereof the greatest being marked with E. is the square of A.D. The next, which is marked with G, is the square of halfe the line, that is, of A.C, And the other two little squares marked with F. and H, be both of one bignes, by reason that I did diuide C.B. into two equall partes, so that you amy take the square F, for the square of D.B, and the square H, for the square of C.D. Now I thinke you doubt not, but that the square E. and the square F, ar double so much as the square G. and the square H, which thing the easyer is to be vnderstande, bicause that the greate square hath in his side iij. quarters of the firste line, which multiplied by itselfe maketh nyne quarters, and the square F. containeth but one quarter, so that bothe doo make tenne quarters.

Then G. contayneth iiij. quarters, seynge his side containeth twoo, and H. containeth but one quarter, whiche both make but fiue quarters, and that is but halfe of tenne.

Whereby you may easylye coniecture, that the meanynge of the theoreme is verified in the figures of this example.

_The xliiij. Theoreme._

If a right line be deuided into ij. partes equally, and an other portion of a righte lyne annexed to that firste line, the square of this whole line so compounded, and the square of the portion that is annexed, ar doule as much as the square of the halfe of the firste line, and the square of the other halfe ioyned in one with the annexed portion, as one whole line.

_Example._