The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 20

In this circle A.B.C.D, I haue drawen first the diameter, whiche is A.D, whiche pa.s.seth (as it must) by the centre E, Then haue I drawen ij. other lines as M.N, whiche is neerer the centre, and F.G, that is farther from the centre. The fourth line also on the other side of the diameter, that is B.C, is neerer to the centre then the line F.G, for it is of lyke distance as is the lyne M.N. Nowe saie I, that A.D, beyng the diameter, is the longest of all those lynes, and also of any other that maie be drawen within that circle, And the other line M.N, is longer then F.G. Also the line F.G, is shorter then the line B.C, for because it is farther from the centre then is the lyne B.C. And thus maie you iudge of al lines drawen in any circle, how to know the proportion of their length, by the proportion of their distance, and contrary waies, howe to discerne the proportion of their distance by their lengthes, if you knowe the proportion of their length. And to speake of it by the waie, it is a maruaylouse thyng to consider, that a man maie knowe an exacte proportion betwene two thynges, and yet can not name nor attayne the precise quant.i.tee of those two thynges, As for exaumple, If two squares be sette foorthe, whereof the one containeth in it fiue square feete, and the other contayneth fiue and fortie foote, of like square feete, I am not able to tell, no nor yet anye manne liuyng, what is the precyse measure of the sides of any of those .ij. squares, and yet I can proue by vnfallible reason, that their sides be in a triple proportion, that is to saie, that the side of the greater square (whiche containeth .xlv. foote) is three tymes so long iuste as the side of the lesser square, that includeth but fiue foote. But this seemeth to be spoken out of ceason in this place, therfore I will omitte it now, reseruyng the exacter declaration therof to a more conuenient place and time, and will procede with the residew of the Theoremes appointed for this boke.

_The .lxi. Theoreme._

If a right line be drawen at any end of a diameter in perpendicular forme, and do make a right angle with the diameter, that right line shall light without the circle, and yet so iointly knitte to it, that it is not possible to draw any other right line betwene that saide line and the circ.u.mfer?ce of the circle. And the angle that is made in the semicircle is greater then any sharpe angle that may be made of right lines, but the other angle without, is lesser then any that can be made of right lines.

_Example._

[Ill.u.s.tration]

In this circle A.B.C, the diameter is A.C, the perpendicular line, which maketh a right angle with the diameter, is C.A, whiche line falleth without the circle, and yet ioyneth so exactly vnto it, that it is not possible to draw an other right line betwene the circ.u.mference of the circle and it, whiche thyng is so plainly seene of the eye, that it needeth no farther declaracion. For euery man wil easily consent, that betwene the croked line A.F, (whiche is a parte of the circ.u.mfer?ce of the circle) and A.E (which is the said perp?dicular line) there can none other line bee drawen in that place where they make the angle. Nowe for the residue of the theoreme. The angle D.A.B, which is made in the semicircle, is greater then anye sharpe angle that may bee made of ryghte lines. and yet is it a sharpe angle also, in as much as it is lesser then a right angle, which is the angle E.A.D, and the residue of that right angle, which lieth without the circle, that is to saye, E.A.B, is lesser then any sharpe angle that can be made of right lines also. For as it was before rehersed, there canne no right line be drawen to the angle, betwene the circ.u.mference and the right line E.A. Then must it needes folow, that there can be made no lesser angle of righte lines. And againe, if ther canne be no lesser then the one, then doth it sone appear, that there canne be no greater then the other, for they twoo doo make the whole right angle, so that if anye corner coulde be made greater then the one parte, then shoulde the residue bee lesser then the other parte, so that other bothe partes muste be false, or els bothe graunted to be true.

_The lxij. Theoreme._

If a right line doo touche a circle, and an other right line drawen frome the centre of the circle to the pointe where they touche, that line whiche is drawenne frome the centre, shall be a perpendicular line to the touch line.

_Example._

[Ill.u.s.tration]

The circle is A.B.C, and his centre is F. The touche line is D.E, and the point wher they touch is C. Now by reason that a right line is drawen frome the centre F. vnto C, which is the point of the touche, therefore saith the theoreme, that the sayde line F.C, muste needes bee a perpendicular line vnto the touche line D.E.

_The lxiij. Theoreme._

If a righte line doo touche a circle, and an other right line be drawen from the pointe of their touchinge, so that it doo make righte corners with the touche line, then shal the centre of the circle bee in that same line, so drawen.

_Example._

[Ill.u.s.tration]

The circle is A.B.C, and the centre of it is G. The touche line is D.C.E, and the pointe where it toucheth, is C. Nowe it appeareth manifest, that if a righte line be drawen from the pointe where the touch line doth ioine with the circle, and that the said lyne doo make righte corners with the touche line, then muste it needes go by the centre of the circle, and then consequently it must haue the sayde c?tre in him. For if the saide line shoulde go beside the centre, as F.C. doth, then dothe it not make righte angles with the touche line, which in the theoreme is supposed.

_The lxiiij. Theoreme._

If an angle be made on the centre of a circle, and an other angle made on the circ.u.mference of the same circle, and their grounde line be one common portion of the circ.u.mference, then is the angle on the centre twise so great as the other angle on the circufer?ce.

_Example._

[Ill.u.s.tration]

The circle is A.B.C.D, and his centre is E: the angle on the centre is C.E.D, and the angle on the circ.u.mference is C.A.D t their commen ground line, is C.F.D. Now say I that the angle C.E.D, whiche is on the centre, is twise so greate as the angle C.A.D, which is on the circ.u.mference.

_The lxv. Theoreme._

Those angles whiche be made in one cantle of a circle, must needes be equal togither.

_Example._

Before I declare this theoreme by example, it shall bee needefull to declare, what is it to be vnderstande by the wordes in this theoreme. For the sentence canne not be knowen, onles the uery meaning of the wordes be firste vnderstand. Therefore when it speaketh of angles made in one cantle of a circle, it is this to be vnderstand, that the angle muste touch the circ.u.mference: and the lines that doo inclose that angle, muste be drawen to the extremities of that line, which maketh the cantle of the circle. So that if any angle do not touch the circ.u.mference, or if the lines that inclose that angle, doo not ende in the extremities of the corde line, but ende other in some other part of the said corde, or in the circ.u.mference, or that any one of them do so eande, then is not that angle accompted to be drawen in the said cantle of the circle. And this promised, nowe will I c.u.mme to the meaninge of the theoreme. I sette forthe a circle whiche is A.B.C.D, and his centre E, in this circle I drawe a line D.C, whereby there ar made two cantels, a more and a lesser. The lesser is D.E.C, and the geater is D.A.B.C. In this greater cantle I drawe two angles, the firste is D.A.C, and the second is D.B.C which two angles by reason they are made bothe in one cantle of a circle (that is the cantle D.A.B.C) therefore are they both equall. Now doth there appere an other triangle, whose angle lighteth on the centre of the circle, and that triangle is D.E.C, whose angle is double to the other angles, as is declared in the lxiiij.

Theoreme, whiche maie stande well enough with this Theoreme, for it is not made in this cantle of the circle, as the other are, by reason that his angle doth not light in the circ.u.mference of the circle, but on the centre of it.

[Ill.u.s.tration]

_The .lxvi. theoreme._

Euerie figure of foure sides, drawen in a circle, hath his two contrarie angles equall vnto two right angles.

_Example._

[Ill.u.s.tration]

The circle is A.B.C.D, and the figure of foure sides in it, is made of the sides B.C, and C.D, and D.A, and A.B. Now if you take any two angles that be contrary, as the angle by A, and the angle by C, I saie that those .ij. be equall to .ij. right angles. Also if you take the angle by B, and the angle by D, whiche two are also contray, those two angles are like waies equall to two right angles. But if any man will take the angle by A, with the angle by B, or D, they can not be accompted contrary, no more is not the angle by C. estemed contray to the angle by B, or yet to the angle by D, for they onely be accompted _contrary angles_, whiche haue no one line common to them bothe. Suche is the angle by A, in respect of the angle by C, for there both lynes be distinct, where as the angle by A, and the angle by D, haue one common line A.D, and therfore can not be accompted contrary angles, So the angle by D, and the angle by C, haue D.C, as a common line, and therefore be not contrary angles. And this maie you iudge of the residewe, by like reason.

_The lxvij. Theoreme._

Vpon one right lyne there can not be made two cantles of circles, like and vnequall, and drawen towarde one parte.

_Example._

[Ill.u.s.tration]

Cantles of circles be then called like, when the angles that are made in them be equall. But now for the Theoreme, let the right line be A.E.C, on whiche I draw a cantle of a circle, whiche is A.B.C. Now saieth the Theoreme, that it is not possible to draw an other cantle of a circle, whiche shall be vnequall vnto this first cantle, that is to say, other greatter or lesser then it, and yet be lyke it also, that is to say, that the angle in the one shall be equall to the angle in the other. For as in this example you see a lesser cantle drawen also, that is A.D.C, so if an angle were made in it, that angle would be greatter then the angle made in the cantle A.B.C, and therfore can not they be called lyke cantels, but and if any other cantle were made greater then the first, then would the angle in it be lesser then that in the firste, and so nother a lesser nother a greater cantle can be made vpon one line with an other, but it will be vnlike to it also.

_The .lxviij. Theoreme._

Lyke cantelles of circles made on equal righte lynes, are equall together.

_Example._

What is ment by like cantles you haue heard before. and it is easie to vnderstand, that suche figures a called equall, that be of one bygnesse, so that the one is nother greater nother lesser then the other. And in this kinde of comparison, they must so agree, that if the one be layed on the other, they shall exactly agree in all their boundes, so that nother shall excede other.

[Ill.u.s.tration]

Nowe for the example of the Theoreme, I haue set forthe diuers varieties of cantles of circles, amongest which the first and seconde are made vp equall lines, and ar also both equall and like. The third couple ar ioyned in one, and be nother equall, nother like, but expressyng an absurde deformitee, whiche would folowe if this Theoreme wer not true. And so in the fourth couple you maie see, that because they are not equall cantles, therfore can not they be like cantles, for necessarily it goeth together, that all cantles of circles made vpon equall right lines, if they be like they must be equall also.

_The lxix. Theoreme._

In equall circles, suche angles as be equall are made vpon equall arch lines of the circ.u.mference, whether the angle light on the circ.u.mference, or on the centre.

_Example._