The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 18

[Ill.u.s.tration]

The line is A.B, and is diuided firste into twoo equal partes in C, and th? is there annexed to it an other portion whiche is B.D. Now saith the Theoreme, that the square of A.D, and the square of B.D, ar double to the square of A.C, and to the square of C.D. The line A.B. ctaining four partes, then must needes his halfe containe ij. partes of such partes I suppose B.D.

(which is the nexed line) to containe thre, so shal the hole line cprehend vij. parts, and his square xlix. parts, where vnto if you ad y^e square of the annexed lyne, whiche maketh nyne, than those bothe doo yelde, lviij. whyche must be double to the square of the halfe lyne with the annexed portion. The halfe lyne by it selfe conteyneth but .ij. partes, and therfore his square dooth make foure. The halfe lyne with the annexed portion conteyneth fiue, and the square of it is .xxv, now put foure to .xxv, and it maketh iust .xxix, the euen halfe of fifty and eight, wherby appereth the truthe of the theoreme.

_The .xlv. theoreme._

In all triangles that haue a blunt angle, the square of the side that lieth against the blunt angle, is greater than the two squares of the other twoo sydes, by twise as muche as is comprehended of the one of those .ij. sides (inclosyng the blunt corner) and the portion of the same line, beyng drawen foorth in lengthe, which lieth betwene the said blunt corner and a perpendicular line lightyng on it, and drawen from one of the sharpe angles of the foresayd triangle.

_Example._

For the declaration of this theoreme and the next also, whose vse are wonderfull in the practise of Geometrie, and in measuryng especially, it shall be nedefull to declare that euery triangle that hath no ryght angle as those whyche are called (as in the boke of practise is declared) sharp cornered triangles, and blunt cornered triangles, yet may they be brought to haue a ryght angle, eyther by partyng them into two lesser triangles, or els by addyng an other triangle vnto them, whiche may be a great helpe for the ayde of measuryng, as more largely shall be sette foorthe in the boke of measuryng. But for this present place, this forme wyll I vse, (whiche Theon also vseth) to adde one triangle vnto an other, to bryng the blunt cornered triangle into a ryght angled triangle, whereby the proportion of the squares of the sides in suche a blunt cornered triangle may the better bee knowen.

[Ill.u.s.tration]

Fyrst therfore I sette foorth the triangle A.B.C, whose corner by C. is a blunt corner as you maye well iudge, than to make an other triangle of yt with a ryght angle, I must drawe forth the side B.C. vnto D, and fr the sharp corner by A. I brynge a plumbe lyne or perp?dicular on D. And so is there nowe a newe triangle A.B.D. whose angle by D. is a right angle. Nowe accordyng to the meanyng of the Theoreme, I saie, that in the first triangle A.B.C, because it hath a blunt corner at C, the square of the line A.B. whiche lieth against the said blunte corner, is more then the square of the line A.C, and also of the lyne B.C, (whiche inclose the blunte corner) by as muche as will amount twise of the line B.C, and that portion D.C. whiche lieth betwene the blunt angle by C, and the perpendicular line A.D.

The square of the line A.B, is the great square marked with E.

The square of A.C, is the meane square marked with F. The square of B.C, is the least square marked with G. And the long square marked with K, is sette in steede of two squares made of B.C, and C.D. For as the shorter side is the iuste lengthe of C.D, so the other longer side is iust twise so longe as B.C, Wherfore I saie now accordyng to the Theoreme, that the greatte square E, is more then the other two squares F. and G, by the quant.i.tee of the longe square K, wherof I reserue the profe to a more conuenient place, where I will also teache the reason howe to fynde the lengthe of all suche perpendicular lynes, and also of the line that is drawen betweene the blunte angle and the perpendicular line, with sundrie other very pleasant conclusions.

_The .xlvi. Theoreme._

In sharpe cornered triangles, the square of anie side that lieth against a sharpe corner, is lesser then the two squares of the other two sides, by as muche as is comprised twise in the long square of that side, on whiche the perpendicular line falleth, and the portion of that same line, liyng betweene the perpendicular, and the foresaid sharpe corner.

_Example._

[Ill.u.s.tration]

Fyrst I sette foorth the triangle A.B.C, and in yt I draw a plube line from the angle C. vnto the line A.B, and it lighteth in D. Nowe by the theoreme the square of B.C. is not so muche as the square of the other two sydes, that of B.A. and of A.C. by as muche as is twise conteyned in the lg square made of A.B, and A.D, A.B. beyng the line or syde on which the perpendicular line falleth, and A.D. beeyng that portion of the same line whiche doth lye betwene the perpendicular line, and the sayd sharpe angle limitted, whiche angle is by A.

For declaration of the figures, the square marked with E. is the square of B.C, whiche is the syde that lieth agaynst the sharpe angle, the square marked with G. is the square of A.B, and the square marked with F. is the square of A.C, and the two longe squares marked with H.K, are made of the hole line A.B, and one of his portions A.D. And truthe it is that the square E. is lesser than the other two squares C. and F. by the quant.i.tee of those two long squares H. and K. Wherby you may consyder agayn, an other proportion of equalitee, that is to saye, that the square E. with the twoo longsquares H.K, are iuste equall to the other twoo squares C. and F. And so maye you make, as it were an other theoreme. _That in al sharpe cornered triangles, where a perpendicular line is drawen frome one angle to the side that lyeth againste it, the square of anye one side, with the ij.

longesquares made at that hole line, whereon the perpendicular line doth lighte, and of that portion of it, which ioyneth to that side whose square is all ready taken, those thre figures, I say, are equall to the ij. squares, of the other ij. sides of the triangle._ In whiche you muste vnderstand, that the side on which the perpendiculare falleth, is thrise vsed, yet is his square but ones mencioned, for twise he is taken for one side of the two long squares. And as I haue thus made as it were an other theoreme out of this fourty and sixe theoreme, so mighte I out of it, and the other that goeth nexte before, make as manny as woulde suffice for a whole booke, so that when they shall bee applyed to practise, and consequently to expresse their benefite, no manne that hathe not well wayde their wonderfull commoditee, would credite the possibilitie of their wonderfull vse, and large ayde in knowledge. But all this wyll I remitte to a place conuenient.

_The xlvij. Theoreme._

If ij. points be marked in the circ.u.mfer?ce of a circle, and a right line drawen frome the one to the other, that line must needes fal within the circle.

_Example._

[Ill.u.s.tration]

The circle is A.B.C.D, the ij. poinctes are A.B, the righte line that is drawenne frome the one to the other, is the line A.B, which as you see, must needes lyghte within the circle. So if you putte the pointes to be A.D, or D.C, or A.C, other B.C, or B.D, in any of these cases you see, that the line that is drawen from the one p.r.i.c.ke to the other dothe euermore run within the edge of the circle, els canne it be no right line. How be it, that a croked line, especially being more croked then the portion of the circ.u.mference, maye bee drawen from pointe to pointe withoute the circle. But the theoreme speaketh only of right lines, and not of croked lines.

_The xlviij. Theoreme._

If a righte line pa.s.singe by the centre of a circle, doo crosse an other right line within the same circle, pa.s.singe beside the centre, if he deuide the saide line into twoo equal partes, then doo they make all their angles righte.

And contrarie waies, if they make all their angles righte, then doth the longer line cutte the shorter in twoo partes.

_Example._

[Ill.u.s.tration]

The circle is A.B.C.D, the line that pa.s.seth by the centre, is A.E.C, the line that goeth beside the centre is D.B. Nowe saye I, that the line A.E.C, dothe cutte that other line D.B. into twoo iuste partes, and therefore all their four angles ar righte angles. And contrarye wayes, bicause all their angles are righte angles, therfore it muste be true, that the greater cutteth the lesser into two equal partes, accordinge as the Theoreme would.

_The xlix. Theoreme._

If twoo right lines drawen in a circle doo crosse one an other, and doo not pa.s.se by the centre, euery of them dothe not deuide the other into equall partions.

_Example._

[Ill.u.s.tration]

The circle is A.B.C.D, and the centre is E, the one line A.C, and the other is B.D, which two lines crosse one an other, but yet they go not by the centre, wherefore accordinge to the woordes of the theoreme, eche of theim doth cuytte the other into equall portions. For as you may easily iudge, A.C. hath one porti lger and an other shorter, and so like wise B.D.

Howbeit, it is not so to be vnderstd, but one of them may be deuided into ij. eu? parts, but bothe to bee cutte equally in the middle, is not possible, onles both pa.s.se through the c?tre, therfore much rather wh? bothe go beside the centre, it can not be that eche of theym shoulde be iustely parted into ij. euen partes.

_The L. Theoreme._

If two circles crosse and cut one an other, then haue not they both one centre.

_Example._

[Ill.u.s.tration]

This theoreme seemeth of it selfe so manifest, that it neadeth nother demonstration nother declaraci. Yet for the plaine vnderstanding of it, I haue sette forthe a figure here, where ij. circles be draw?, so that one of them doth crosse the other (as you see) in the pointes B. and G, and their centres appear at the firste sighte to bee diuers. For the centre of the one is F, and the centre of the other is E, which diffre as farre asondre as the edges of the circles, where they bee most distaunte in sonder.

_The Li. Theoreme._

If two circles be so drawen, that one of them do touche the other, then haue they not one centre.

_Example._

[Ill.u.s.tration]

There are two circles made, as you see, the one is A.B.C, and hath his centre by G, the other is B.D.E, and his centre is by F, so that it is easy enough to perceaue that their centres doe dyffer as muche a sonder, as the halfe diameter of the greater circle is lger then the half diameter of the lesser circle. And so must it needes be thought and said of all other circles in lyke kinde.

_The .lij. theoreme._

If a certaine pointe be a.s.signed in the diameter of a circle, distant from the centre of the said circle, and from that pointe diuerse lynes drawen to the edge and circ.u.mference of the same circle, the longest line is that whiche pa.s.seth by the centre, and the shortest is the residew of the same line. And of al the other lines that is euer the greatest, that is nighest to the line, which pa.s.seth by the centre. And ctrary waies, that is the shortest, that is farthest from it. And amongest th? all there can be but onely .ij. equall together, and they must nedes be so placed, that the shortest line shall be in the iust middle betwixte them.

_Example._