Proportion quantities whatever, 
of Figures™ of A, that Am X ; 
—_—\~— 
Definitions. 
Fig. 87. 
Fig. 96. 
Fig. 94. 
Fig. 95. 
Fig. 88. 
214 
and suppose that X is such a part 
then, if denote the number of 
times that X can be taken from B, and D the remain- 
der, ‘we have B=n X4-D, and B—D=nX; and be- 
cause mi nz:rm Xin &,’ therefore m:n:: A: B—D. 
Now as D is less than X, by taking X sufficiently small, 
D may be less than‘any proposed quantity, and B—D 
may differ from B by less than any given quantity ; 
therefore such values may be given to m and n, as shall 
make the ratio of m to as near to the ratio of A to B 
as we please. Hence we may, with perfect confidence, 
apply whatever has been: delivered in this Section con- 
cerning commensurable quantities to such as are incom- 
mensurable. 
SECTION IV. 
Tur Proportion oF FicureEs, 
Definitions. 
1. Equivalent figures are such as have equal surfaces. 
Two figures may be equivalent, although dissimilar. 
For example, a circle may be equal toa square ; a tri- 
angle to a rectangle, &c. 
We shall apply the term equal to such figures only as 
would coincide entirely, if placed the one upon the other. 
2. Two figures are similar, when the angles of the one 
are equal to the angles of the other, each to each, and the 
homologous sides ed an Bythe homologous sides, 
we mean those that have the same position in the two 
figures, or which are adjacent to on angles: the an- 
gles themselves may be called homologous angles. 
8. In two circles, similar sectors, similar arcs, similar 
segments, are those which have equal angles at the cen- 
tre. Thus, ifthe angle A=O, the are BC is similar to 
the are DE, and the sector ABC to the sector ODE 
(Fig. 87.) 
4. The altitude of a triangle ABC, (Fig. 96,) is a 
perpendicular drawn from any one of its angles A upon 
the opposite side BC its base. 
The altitude of a parallelogram ABCE, (Fig. 94,) is 
the distance AD between any two of its parallel sides. 
The altitude of a trapezoid ABCD, is the distance 
EF between its parallel sides. (Fig. 95.) 
5. The area and the surface of a figure, are terms of 
nearly the same import. The area, however, is more 
particularly the quantity of superficies, as expressed by 
some other superficies taken a certain number of times. 
Prop. I. Tueror. 
Parallelograms which have equal bases and equal al- 
titudes are equivalent. 
Let AB be the common base of the two parallelo- 
grams ABCD, ABEF; since they are supposed to 
have the same altitude, their sides DC, FE, opposite to 
their bases, will be in the same straight line parallel to 
AB. But by the nature of parallelograms AD=BC, 
and AF=BE ; also DC=AB, and FE=AB, (26. Bi 
and therefore DC=FE ; and taking away DC and F 
from the same straight line DE, there remains DF=CE: 
Hence the triangles DAT’, CBE have the three sides 
of the one equal to the three sides of the other, each to 
each, therefore they are equal (11. 1.): Now if the 
former be taken away from the quadrilateral ABED, 
there will remain the Bo pseein the AFEB; and if the 
Jatter-be taken from the same quadrilateral, the paral- 
GEOMETRY. 
lelogram ABCD will remain ; therefore the parallelo= Prop 
ABCD is equivalent to the parallelogram ABEF. of Fi 
‘Cor. Every logram ABCD is equal to a ree= 
tangle FBCE of the same base and altitude: (Fig. 89.) Fig. 
Prop. II. | Tittor, | 
Any triangle ABC is half of a parallelogram” ABCD rig, 
of the sat thane anndielii lit’ Pray Ser Tea be a 
For the triangles ABC, ACD are equal, (26. 1.) 
Cor. 1. Therefore a tri ‘ABC is half ofa rec 
tangle BCEF, which has the same base BC and the 
same altitude AO. wet tol ah 
Cor. 2. Triangles which have equal bases and equal 
altitudes are equivalent. ; : ; 
Prop. III. 
Two rectangles of the same altitude are to one ano- 
THEOR. 
ther as their bases. xb 
Let ABCD, AEFD be two rectangles, which have 
a common altitude AD; they are to one another as 
their bases AB, AE. Si'es wae d 
For suppose that the base AB contains seven such Fig 
as the base AE contains four; then, if AB be 
divided into seven equal parts, AE will contain four of 
them. At each point of division draw a dicular 
to the base ; these will form seven equal rectangles (1.) ; 
and as AB contains seven such parts as AE contains 
four, the rectangle AC will also contain seven such 
arts as the rectangle AF contains four; therefore AB 
has to AE the same ratio that the rectangle AC has to 
the rectangle AF. : 
Prop. IV. Tuxor. 
Any two rectangles are to one another as the 
ducts of the numbers which express their bases ood al- 
titudes. 
Let ABCD, AEGF be two rectangles, and let some Fig. 
line taken as an unit be contained m times in AB the 
base of the one, and x times in AD its altitude; 
p times in AE the base of the other, and q times in A 
its altitude ; the rectangle ABCD shall be to the rect. 
angle AEGF as the product mn to the product p q. 
Let the rectangles be so placed, that their bases KB, 
AE may be in a straight line, then their other sides 
AD, AF shall also form a ory line (3. 1.) _Com- 
lete the rectangle EADH, and because this rectangle 
as the same altitude as the rectangle ABCD when EA, 
AB are taken as their bases, and the same altitude as _ 
the rectangle AEGF when AD, AF are taken astheir — 
bases, we have ; 
ABCD: ADHE:: AB: AE:: m:p (3.) ~ 
but m:p::mn:pn (1, 3.) 
therefore ABCD: AEHD:: mn:pn. 
In like manner, it appears that j 
AEHD: AEGF:: AD: AF::2: 
en eee eee 
i1pn: pq. 
Therefore, ex equo, ABCD: AEG ee 
Scnouium. If ABCD, one of the rectangles, (Fig. 
92.) be a square having. the measuring unit for its side, * 
this square may be taken as the measuring unit of sur- 
faces; and because the linear unit AB. is contained p 
times in EF, and g times in EH, by the proposition, _ 
1X1:p ABCD : EFGI ; ee 
hence the rectangle EFGH will contain the superficial 
a 
