OTHERWISE. Plate VII. Fig. 44.

Conftr. To the parallel right-lines given in position, AB and CD, let there be drawn from the point E the right-line FEG: I say, that the ratio of GE to EF is given.

Demonftr. For from the point E let there be drawn to CD the perpendicular EH, and produced to the point K; feeing therefore that from the point E to the rightline CD, given in position, there is drawn the line EH, making the given angle EHG, (a) the faid line EH is given in position. But each line AB and CD is also given in pofition: Therefore (b) each point of interfection H and K is given. But the point E is also given, therefore (c) each of the lines EH and IK is given in Magnitude; and therefore (d) the ratio of the said EH to EK is given. But (e) as EH is to FK so is EG to EF (for the oppofite angles at the point E being equal, and the lines AB and CD parallels, the triangles EHG and EKF are equiangled; and therefore as EH is to EG, so is EK to EF; and by permutation as EH to EK, so is FG to EF.) Therefore the ratio of the faid lines EG to EF is given.

PROP. XXXV. Fig. 45.

If from a given point A, to a right-line BC, given in position, there be drawn a right-line AD, which let be divided in E, in a given ratio (to wit) as AE to ED, and that by the point of fection E there be drawn a right-line FEG, opposite to the right-line BC, given in position, the line FG so drawn shall be given in position.

Constr. For from the point A, let there be drawn the line AH, perpendicular to the line BC.

Demonstr. For feeing that from the given point A there is drawn to BC given in position, the right-line AH making the given angle AHD, (a) the faid line AH is given in position. But BC is also given in pofition: Therefore (6) the point H is given. But the point A is alfo given: Therefore (c) the line AH is given in magnitude and in position. And feeing that (d) as AE is to ED, so is AK to KH, and that the ratio of AE to ED is given, also the ratio of AK to KH is given; and by compounding, (e) the ratio of AH to AK is given. But AH is given in Magnitude: Therefore (f) alfo

AK

AK is given in Magnitude. But AK is also given in pofition, and the point A is given: Therefore (g) the pointg 27. prop. K is also given, and feeing that by the faid given point K there is drawn the line FG, oppofite to the right-line BC given in position; the faid line FG (b) is given in position. h 28. prop.

PROP. XXXVI. Plate VII. Fig. 46.

If from a given point A, there be drawn to a rightline BC given in position, a right-line AD, and to it be added a right-line AE, having to the fame AD a given ratio, and that through the extremity E of the added line AB, there be drawn a right-line FEK, opposite to the line BC, given in position, that fame line FEK spall be given in position.

Constr. For from the point A let there be drawn to the line BC, the perpendicular AL, and let it be prolonged to the point G.

a 30. prop.

c 26. prop.

d 4. 6.

Demonstr. Forasmuch as from the given point A, there is drawn to the right-line BC, given in position, the right-line GL, which makes the given angle GLD, (a) that line GL is given in position. But BC is also given in pofition, therefore (b) the point L is given; and feeing b 26. prop. that the point A is also given, the line (c) AL is given. But forasmuch as the ratio of AE to AD, is given, and that (d) as the said AE is to AD, so is AG to AL; (because the triangles ALD and AGE are equiangled) the ratio of AG to AL is also given, But AL is given in Magnitude: Therefore (e) AG is given in Magnitude. But it is also given in position, and the point A is given : Therefore (f) the point G is also given. And seeing that by the fame given point G there is drawn the line FK, opposite to the right-line BC, given in position, (g) the faid line FK is given in position.

PROP. XXXVII. Fig. 47.

If unto parallel right-lines AB and CD, given in pofition, there be drawn a right-line EF, divided in the point G, in a given ratio, (to wit,) of EG to GF: and if through the point of section G, there be drawn oppofite to the right-lines AB or CD, given in position, a right-line HGK, that line shall be given in pofition.

e 2. prop.

f 27. prop.

§ 28. prop.

2 30. prop.

b 25. prop.

c 26. prop.

Conftr. For let there be taken in the line AB the given point L, and from that point let there be drawn the line LN, perpendicular to CD.

Demonftr. Seeing that from the given point L, there is drawn to the right-line CD, the line LN, making the given angle LND, the said LN (a) is given in position. But CD is also given in position: Therefore the point N (b) is given. But the point L is also given: Therefore (c) the line LN is given; and feeing that the ratio of FG to GE is given, and that * as FG is to GE, so is NM to to ML, the ratio of the said NM to ML is given; and d 6. prop. by compounding, (d) the ratio of LN to LM is also given. But LN is given in Magnitude; therefore ML is (e) given in Magnitude. But it is also given in position, and the point L is given; Therefore the point M tion, and the point Ind (f) is also given. And confidering that through the faid point M there is drawn the right-line KH, opposite to the right-line CD, given in position, the said line KH

e 2. prop.

f 27. prop.

87.5. A 17.5.

is also given in position.

Scholium.

* EUCLIDE supposeth here, that as FG is to GE, so NM is to ML; but by another it is thus demonstrated.

The lines EF and LN are parallels or not parallels : Let them in the first place be parallels, and forafmuch as by Construction the lines EL, FN, EF, and LN, are parallels, EN shall be a parallelogram, and therefore the fide EF is equal to the fide LN. Again, seeing that MG is parallel to NF, and GF to MN, GN shall be also a parallelogram; and therefore the fide GF is equal to the fide MN. Wherefore the equal fides EF and LN, shall have to the equal fides FG and MN, (g) one and the fame ratio. Therefore as EF is to FG, so is LN to MN; and by dividing, (h) as GE to GF, so is LM to MN.

Now suppose that the lines EF and LN (Fig. 48.) are not parallels, but that they meet in the point O. Forafmuch as in the triangle OFN there is drawn ΗΚ, parallel to FN one of the fides; (i) the fides OF and ON are divided proportionally; and therefore as FG is to GO, So is NM to MO. Again, seeing that in the triangle OGM there is drawn EL, parallel to the side GM, the fides OG and OM are divided proportionally : Wherefore (k) as OE is to EG, so is OL to LM, and by compounding, (1) as OG is to EG, so is OM to LM; but it hath been demonstrated that as FG is to GO, So is NM to MO; therefore in ratio of equality, (m) as FG is to GE, so is NM to ML.

PROP. XXXVIII, Plate VII. Fig 49.

If unto parallel right-lines AB and CD, there be drawn a right-line EF, and that to it there be added some other right-line EG, which hath a given ratio to the same EF; and if through the extremity G of the added line EG, there be drawn a right-line HK, against the parallels given in pofition AB and CD, the line drawn HK shall be also given in position.

Conftr. For let there be taken in the line AB, the given point N, and from thence let there be drawn to CD the perpendicular NM, and let it be prolonged to the point L.

Demonßr. Forasmuch as from the given point N there is drawn to the right-line CD, given in position, the right-line NM making a given angle NMF, the faid angle NMF (a) is given in position. But the line CD is also given in position: Therefore (6) the point M is given. But the point N is also given: Therefore (c) the line NM is given, and therefore the ratio of EG to EF is given; and because (d) as EG to EF, so is LN to MN, the ratio of LN to NM is also given: But NM is given, therefore LN is (e) also given. But the point N is given: Therefore (f) the point L is also given. Seeing then that by the given point L there is drawn the rightline HK, opposite to the line AB given in position, (g) the faid line AK is also given in pofition.

PROP. XXXIX. Fig. 50.

If all the fides of a triangle ABC are given in magnitude, the triangle is given in kind.

Conftr. For, let there be exposed the right-line DG given in position, ending in the point D; but being infinite towards the other part G, and therein let be taken DE, equal to AB.

Demonstr. Now seeing the said AB is given in magnitude, DE is so also; but the fame DE is also given in is position, and the point D is given: Therefore (a) the point E is given.

Again, Let EF be put equal to BC; and feeing that BC is given in magnitude, EF shall be so also. But the faid EF is in like manner given in position, and the point E is given: Therefore (6) the point F is given.

a 30. prop. b 25. prop. c 26. prop. 26.

d fch. 37.

prop.
e 2. prop.
f 27. prop.

g 28. prop.

a 27. prop.

b 27. prop

Furthermore, Let FG be taken equal to AC. Now forasmuch as the faid AC is given in magnitude, FG is

c 6. def.

d 6. def.

e 25. prop.

f 26. prop.

g 1. prop.

h fch. 30. prop.

i 15. def.

fo alfo. But FG is also given in position, and the point F is given: Therefore the point G is also given. Now from the center E, with the distance ED, let there be described the circle DHK, (c) and that circle shall be given in position. Again, on the center F, and distance FG, let there be described the circle GLK. Therefore (d) the faid circle GLK is given in position; and therefore (e) the point of Interfection K is given. But each of the points E and F is given: Therefore each line (f) EK, EF, and FK, is given in position and magnitude. Therefore the triangle FK is given * in kind; but it is equal and alike to the triangle ABC; and therefore the triangle ABC is also given in kind.

Scholium.

* EUCLIDE supposeth here, that a triangle whose fides are given in magnitude and position, is given in kind, but the antient Interpreters demonstrated it in a manner thus. Forafmuch as the right-lines KE and EF are given, (g) the ratio which they have to one another is also given. Also the right-lines EF and FK being given, their ratio is also given; and in like manner, the ratio of the faid EK and FK is given. Again, seeing that the same lines KE and EF are given in position, (h) the angle KEF is given in magnitude: Moreover, the right-lines EF and FK being given in position, the angle EFK is given in magnitude, as is also the residue EKF, and so in the triangle EKF are all the angles given, and also the ratio's of the fides: Therefore (i) the Said triangles EKF is given in kind,

PROP. XL. Plate VII. Fig. 37, 38.

If the angles of a triangle ABC, are given in mag. nitude, the triangle is given in kind.

Conftr. Let there be exposed the right-line DE, given in position and in magnitude; and let there be constitued at the point D the angle EDF, equal to the angle CBA, and at the point E the angle DEF, equal to the angle BCA ; therefore the third angle BAC is equal to the third angle BFE.

Demonstr.