The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

The first Theoreme. The 4. Proposition. If there be two triangles, of which two sides of th'one be equal to two sides of the other, eche side to his correspondent side, and hauing also on angle of the one equal to one angle of the other, namely, that angle which is contayned vnder the equall right lines: the base also of the one shall be equall to the base of the other, and the one triangle shall be equal to the other triangle, and the other angles remayning shal be equall to the other an∣gles remayning, the one to the other, vnder which are subten∣ded equall sides.

SVppose that there be two triangles ABC, & DEF, ha∣uing two sides of the one, namely AB, and AC, equall to two sides of the other, namely, to DE and DF, the one to the other, that is, AB to DE, and AC to DF: hauyng al∣so the angle BAC, equall to the angle EDF. Then I say that the base also BC is equall to y^e base EF: & y^t the tri∣angle ABC, is equall to the triangle DEF: and y^t the o∣ther angles remainyng are equall to the other angles re∣mayning, the one to the other, vnder which are subtended equall sydes: that is, y^t the angle ABC is equall to the angle DEF, and y^t the angle ACB is equall to to the angle DFE. For the triangle ABC ex∣actly

[illustration]

agreyng with the triangle DEF, and the point A being put vpō the point D, & the right line AB vpon the right line DE, the pointe B also shall exactly agree with the pointe E: for that (by supposition) the line AB is equal to the line DE. And the line AB exactly agreeyng with the line DE, the right line also AC exact∣ly agreeth with the right line DF, for that (by supposition) the angle BAC is equall to the an∣gle EDF. And forasmuch as the right line AC is also (by supposition) equall to the right line DF, therfore the pointe C exactly agreeth with the pointe F. A∣gaine forasmuch as the pointe C exactly agreeth with the poynte F, and the point B exactly agreeth with the point E: therefore the base BC shall exactly agree with the base EF. For if the point B do exactly agree with the point E, and the point C with the point F, and the base BC do not exactly agre wyth the base EF, then two right lines do include a superficies: which (by the 10. cōmon sentence) is impossible. VVherfore the base BC exactly agreeth w^t the base EF, and therfore is equall vnto it. VVherfore the whole triangle ABC exactly a∣greeth with the whole triangle DEF, & therfore (by the 8. common sentence) is equall vnto it. And (by the same) the other angles remayning exactly agree with the other angles remayning, and are equall the one to the other: that is, the angle ABC to the angle DEF, and the angle ACB to the angle DFE. If therfore there be two triangles, of which two sides of the one, be equall to two sydes of the other, eche to his correspondent side, and hauing also one angle of the one equall to one angle of the other, namely, that angle which is contayned vn∣der the equall right lines: the base also of the one shall be equall to the base of the other, and the one triangle shall be equall to the other triangle, and the other an∣gles remainyng shall be equall to the other angles remayning, the one to the o∣ther, vnder which are subtended eqaull sydes: whiche thing was required to be demonstrated.

This Proposition which is a Theoreme, hath two things geuen: namely, the

equality of two sides of the one triangle, to two sides of the other triangle, and the equalitie of two angles contayned vnder the equall sydes. In it also are thre thinges required. The equality of base to base: the equality of field to field: and the equality of the other angles of the one triangle to the other angles of the o∣ther triangle, vnder which are subtended equall sides.

One side of a playne figure is equall to an other, and so generally one right lyne is equall to an other, when the one being applied to the other, theyr ex∣treames agree together. For otherwise euery righte line applied to any right lyne, agreeth therwith: but equall right lines only, agree in the extremes.

One rectilined angle is equall to an other rectilined angle, when one of the sides which comprehendeth the one angle, being set vpon one of the sides which comprehendeth the other angle, the other side of the one agreeth with the other syde of the other. And that angle is the greater, whose syde falleth without: and that the lesse, whose syde falleth within.

VVhere as in this proposition is put this particle eche to his correspondent side, (in¦stede wherof often times afterward is vsed this phrase the one to the other) it is of ne∣cessity so put. For otherwise two sydes of one triangle added together, may be e∣quall to two sydes of an other triangle added together, and the angles also con∣tayned vnder the equall sydes may be equall: and yet the two triangles may not∣withstanding be vnequall. VVhere note that a triangle is sayd to be equall to an other triangle, when the field or area of the one is equall to the area of the other. And the area of a triangle, is that space, which is contayned within the sydes of a triangle. And the circuite or compasse of a triangle is a line composed of all the sides of a triangle. And so may you think of all o••her rectilined figures. And now to proue that there may be two triangles, two sydes of one of which being added together, may be equall to two sydes o•• the other added together, and the angles contayned vnder the equall sydes may be equall, and yet notwithstanding the two triangles vnequall. Suppose that there be two rectangle triangles: namely, ABC, and DEF, and let their right angles be BAC and EDF. And in the tri∣angle ABC let the syde AB b•• 3. and the syde AC 4. which both added toge∣ther make 7.

And in the

[illustration]

triāgle DEF, let the side DE be 2. and the side DF be 5. whiche added toge∣ther make al¦so 7. & so the sydes of the one triangle added together, are equall to the sides of the other triā∣gle added together. Yet are both the triangles vnequall, and also their bases. For the area of the triangle ABC is 6 and his base is 5. And the area of the triangle DEF is 5: and his base •• 29. So that to haue the areas of two triangles to be e∣quall, it is requisite that all the sydes of the two triangles be equall, eche to hys correspondent syde. It happeneth also sometymes in triangles, that the areas of them beyng equall, their sydes added together shall be vnequall. And contrari∣wise,

their sides beyng equall, their areas be vnequall. As in these figures here put it is plaine to see. In the first

[illustration]

example the areas of the two triangles are equal, for they are eche 12. and the sides in ech ad∣ded together are vnequall, for in the one triangle the sides ad∣ded together make 18. and in the other they make 16. But in the second exāple the areas of the two triāgles are vnequal, for the one is 12. and th'other is 15. But the sides added to∣gether in eche are equall, for in eche they make 18.

That angle is said to sub∣tend a side of a triāgle, which is placed directly opposite, & against that side. That side also is sayd to subtend an an∣gle, which is opposite to the angle. For euery angle of a triangle is contayned of two sydes of the triangle, and is subtended to the third side.

This is the first Proposition in which is vsed a demonstration leading to an absurditi••, o•• an impossib••litie. VVhich is a demonstration that proueth not di∣rectly the thing entended, by principles, or by thinges before proued by these principles: but proueth the contrary therof to be impossible, & so doth indirect∣ly proue the thing entended.

Notes

• Demonstratio•• lea••ing to an absurditie.
• Two 〈…〉〈…〉
• Three thinges required in it.
• How one side i•• equall to an o∣ther, & so gene¦••ally how one right line is e∣quall to an other.
• How one rectili••ne•• angle is e∣qual to an other
• Why this parti∣cle 〈◊〉〈◊〉 to his c••r••espondent side, is put.
• How one trian∣gle is equal to an other
• What the fielde or area of a tri∣angle is, and so of any rectilined 〈◊〉〈◊〉.
• What 〈…〉〈…〉 of a triangle is, and ••o also of a∣ny 〈◊〉〈◊〉 fi∣gure.
• How an angle is sayd to subtēd a side: and a side an angle.
• This proposition proued by a de∣monstration lea¦ding to an absur¦dity.