of the obliquangular Parallelogram, will Fa be the fame with the Content of a Rectangle fo made.

1

And in like manner, from Corol. 4, of 3. this Theorem, is deduced the Method of The Mea Suring of finding the Content of any Triangle, viz. Triangles, by compleating the Parallelogram, of deduced which the Triangle given is an Half of Paralfrom that and, if the faid Parallelogram be obli- lelograms quangular, by making a Rectangle upon the fame or an equal Bafis, and within the fame Parallels, as is the obliquangular Parallelogram. For, by Corol. 4, it is fhewn, that the Content of a Triangle, is equal to Half the Content of a Parallelogram upon the fame Bafis, and within the fame Parallels.

The Method of Meafuring (or finding: 4 the Content of) any Triangle being thus The Mea difcover'd, thence is deduced the Me- TrapeziJuring of a thod of Meafuring any Trapezium. For um, dedu any Trapezium may be refolv'd into two ced from the Measu Triangles by a Diagonal; the Contents ring of of which two Triangles being found, the Triangles Content of the whole Trapezium is found, being the Sum of the Contents of the two Triangles. Thus, Fig. 73, the Fig. 73 Trapezium ABCD, is by the Diagonal AC refolv'd into two Triangles ABC and ACD. And the Content of the Tri angle ABC, is equal to Half the obli quangular Parallelogram ABHC, (being both

5.

Method

used in

do this.

both on the fame Bafis AC, and within the fame Parallels AC and BH,) and confequently is equal to Half the Rectan gle KBHG (Parallelogram ABHC, as having the fame Bafis BH, and being within the fame Parallels AC or AG, and BH). So the other Triangle ACD= Parallelogram ACDE Parallelogram DEFL. Wherefore, the Content of the whole Trapezium ABCD-ABC+ACD ABHC+ACDE KBHG+DEFL:

X.

But now the Content of a Rectangle The fhort KBHG, (or DEFL) being found by multiplying KB into BH, (or LD into DE ;) Practice to and the Triangle ABC, being only KBHG, (or the Triangle ACD, being only DEFL.) it follows, that the Content of the Triangle ABC, must be equal. to KB BH, or KB BH, (and likewife ACD=LD×÷DE, or to LD-DE). And the Line KB being a Perpendicular let fall from B, the Angle (of the Triangle ABC) oppofite to the Bafis AC, and the faid Bafis AC, being equal to BH, the other contiguous Side of the Rectangle KBHG; hence it comes to pass, that (it is not neceffary in Practice to draw, either the whole obliquangular Parallelogram ABHC, or the whole Rectangle KBHG, which are the Doubles of the Triangle ABC, but) in Practice it is fufficient to know the Length of the Per

pendicular

pendicular KB, in order to know the Content of the Triangle ABC: ForafImuch as KB AC (KC BH) is the Content of the faid Triangle. And in like manner in the Triangle ACD, befides the Length of the Bafis AC, it is fufficient to know the Length of the Perpendicular DL, in order to know the Content of the Triangle ACD. For LD×÷AC(=LD-DE) is the Content of the faid Triangle, as has been already thewn.

Once more, forafmuch as the Diago-, nal AC of the Trapezium ABCD is the The Rules common Bafis, both of the Triangle the Area for finding ABC, and alfo of the Triangle ACD, of a Tra hence is formed this univerfal Rule for pezium. finding the Content of a Trapezium ABCD, viz. the Diagonal AC of the Trapezium ABCD, being multiplied into Half the Sum of the two Perpendiculars DL and KB, in the two Triangles ABC and ACD, (or, which comes to the fame, Half the Diagonal AC being mul tiplied into the whole Sum of the two Perpendiculars KB and DL,) will give the Content of the Trapezium ABCD.

Now there being no confiderable Piece of Land, but what may be reduced to fome of the fore-mention'd Figures, viz. either a Triangle, Parallelogram, or Trapezium; it is evident from what has Q 2

been

7.

Of laving out Land.

been faid, how much the Art of Survey. ing depends on this 7th Theorem, and its four firft Corollaries, in finding out the Measure or fuperficial Content of any given Ground or Piece of Land.

And it is obvious, that on the fame Principles is founded in great Measure, that other Part of Surveying, viz. laying out any Quantity of Land defir'd. And to this the two laft Corollaries of this Theorem are particularly fubfervient. For thereby is demonftrated, how, any Piece of Land (fuppofing it reducible to one or more Parallelograms or Triangles) being given, there may be laid out Ito another Piece of Land, twice or thrice (c) as big; namely, by doubling.or tripling (c.) either the Bafis, or elfe the Altitude of the Parallelograms or Triangles, into which the Piece of Land given is reducible.

Fig. 74.

THEOREM VIII.

A Line DE, drawn parallel to any one Side BC of a Triangle ABC, will cut the other two Sides of the Triangle proportionally, viz. AD: DB:: AE EC.

Demon. Draw the Lines CD and BE. Now the Triangle DEB DEC, by Cor. 2, Theorem 7. Wherefore, by Axiom 6. ADE: DBE :: ADE: DEC. But, by

Corel.

Corol. 6, Theorem 7,ADE : DBE :):)
AD: DB, and by the fame, ADE : DEC
(DBE): AE: EC. Therefore, by
Axiom 9, AD: DB:: AE: EC.

2 USE.

Among feveral other Ufes of this Theorem, it will be fufficient here to observe, that on it is founded the Method of dividing a Line into any Number of Parts requir'd, as alfo of finding a fourth Line proportional to three given, and a third proportional to two given,

THEOREM IX.

Equiangular Triangles ABC and DEF, Fig. 75, have the Sides about the equal Angles & 76, proportional, viz. AB: BC:: DE: EF, and BC: CA :: EF : FD, and CA : AB:: FD: DE.

2

Demon. The Angle E being given equal to the Angle B, upon BC fet off Bf EF, and upon BA fet off Bd=ED, and draw the Line df, which will be equal to DF, by Corol. 1, Theorem 5. Now, by Theorem the 8th, Bd : dA :: Bf: fC. Wherefore, inverfely Ad: dB: Cf fB. Wherefore, by Compofition (AddB, i. e.) AB: dB:: (Cf+fB, i.e.)