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F

GD

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H

A Triangle may be made of them, if any two taken together be greater than the third ; but that the thing is so, make the Ang. HCK = B, and CK=CH, and draw HK, IK. then KH = FG. and because the Ang. KCI d

A; the Line KIDE. But KISHI + HK (FG); therefore DE HI + FG. In like manner, we demonftrate that any two are greater than the third, and consequently a Triangle may be made of them. Q. F. D.

Probl. PRO P. XXIII.

To make a Solid Angle MHIK of three plane Angles A, B, C, whereof any two howsoever taken

M

K

th

fha

KI

to

An the

let

HK cau

fore

fore

MI

whe

DE

HM

=日

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are greater than the third; * but these three Angles * 21. 11. must be less than four right Angles.

Make AD, AE, BE, BF, CF, CG equal to one another, and make the Triang. HKI of the three right Lines HI = DE, IK = EF, KH = FG, about which describe & a Circle LHKI.

f 22. 1.

& 5.4.

In conft, & 8.1. i 21.Ι. 4 cor.

Now AD HL, the Point L being the Centre; for if AD be, or HL, then will the Ang. Ahor HLI. In like manner shall B be =, or - HLK, and C =, or 7 KLI. Whence A + B + C are * either equal k to four right Angles, or else exceed four right 13. 1. Angles, which is contrary to the Hypothefis ; therefore AD - HL.

m

Again, let AD be1 = HL + LM2, and 1 Schol. let LM be perpend. to the Plane of the Circle 47. 1. HKI, and draw HM, KM, IM. Then be- 12. 11. cause the Ang. HLM is a right one; there- 3 def. 11. fore MH =HL+LM2 =

2

2

PAD. There- 47. I. fore MH = AD. By the like Argument MK, P conft. MI, AD (that is, AE, EB, &c.) are equal;

-whence since HM = AD, and MI = AE, and 9 8. 1.
DEP = HI; therefore is the Ang. A =
HMI; and in like manner the Ang. IMK
= B, and the Ang. HMK = C. Whence
the solid Angle at M is made of three given
Plane Angles. Q. E. F.

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II.

I.

=

d

And in like manner, the other Planes of the def. 1. Parallelepipedon are Parallelograms. Therefore since AF is parallel to HG, and AD to HC, the Ang. FAD shall be = CHG; whence because AFd. HG, and AD = HC, 1 and consequently AF: AD :: HG: HC; therefore the Triangles FAD, GAH are f Similar and Equal; and so the Parallelograms AE, HB are Ex. 1. Similar and Equal. And the same may be

5.

.

.

prov'd of the other parallel Planes.

&c.

PROP. XXV.

Therefore,

If a folid Parallelepipedon ABCD be cut by a Plane EF parallel to opposite Planes AD, BC; it shall be as the Base AH is to the Base BH, so is the Solid AHD to the Solid BHC.

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Conceive the Parallelepip. ABCD to be continued out both ways, and take AI = AE, and

Fro the Pl EG, C

the Pla

and A

BK= EB, and put the Planes IQ, KP, parallel Ang. to the Planes AD, BC. The Parallelograms IM, 1. AH, and DL, DG, and * IQ, AD, EF, &c. are Similar and Equal. Whence the Ppp. AQ = AF; and for the fame Reason the Ppp. BPBF. Therefore the Solids IF and EP are

f. 6.

.11.

1

Plane

def.

AL. T

AH: BH:: AF: EC. Q. E. D.

24.11. && 9 def.

may be apply'd to any Prism. Whence

COROL.

Prism be cut by a plane Parallel to
Lanes, the Section shall be a Figure
Similar to the opposite Planes.

PROP. XXVI. Probl.

a folid Angle AHIL equal to a given CDEF, at a Point A, in a given Right

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II.

6 def. 5.

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ny point F, let fall FG perpend. to 11.11. ; and draw the right Lines DF, FE, ,CG. Make AH = CD, and the AI=DCE, and AI = CE ; and in HAI make the Ang. HAK = DCG, = CG; then raise KL perpend. to the AI, and let KL be = GF, and draw en shall the folid Angle AHIL be equal

1

PROP. XXVII.

From a right Line given AB to describe a Parallelepipedon AK Similar, and in like manner fituate to a folid Parallelepipedon given CD.

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a 26. 11.

12.6. €22.5.

f

1 def. 6.

24. 11.

Make a a folid Angle A of the plane Angles BAH, HAI, BAI, which are equal to FCE, ECG, FCG; also make FC: CE :: BA: AH. And CE : CG :: AH : AI, (whence by Equality it shall be FC: CG :: BA: AI) and compleat the Ppp. AK, which shall be fimilar to the given one.

C

For by Constr. the Pgrs. BH, FEd; and HI, EG; and d BI, FG, are similar; and fo the opposite Planes of these to the opposite Planes of those: therefore the fix Planes of the Solid AK are similar to the six Planes of the 9. def. 11. Solid CD, and confequently AK, CD are fsimilar. Q. E. F.

PROP.

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