DEM. ex abs.

5 Concl.

6 Recap.

Thus the less is declared equal to the greater, which is absurd;

.. AB is not AC, that is, AB= AC. Wherefore if two angles of a triangle, &c.

COR-Every equiangular triangle shall also be equilateral.

EXP. |1| Hyp.

2 Concl.

DEM. 1 H. & P. 6. 2 H. & P. 6.

3 Ax. 1.

4 Concl.

5 Recap.

Q.E.D.

Hence the side AB=BC, BC to CA, and

CA to AB.

Wherefore, every equiangular triangle, &c.

Q.E.D.

Geo

SCHOLIUM.-1. This proposition is named the converse of the 5th. metrical conversion takes place when the hypothesis of the former proposition is made the predicate of the latter, and vice versâ, as in Props. 5 and 6, 18 and 19, 24 and 25 of this Book.

2. Converse theorems are not universally true; for instance, the following direct proposition is universally true,"If two triangles have their three sides respectively equal, the three angles of each shall be respectively equal;" but the converse is not universally true, namely,-"If two triangles have the three angles in each respectively equal, the three sides are respectively equal."-Potts' Euclid, p. 48. To the equality of triangles it is always indispensable that one side at least of the one triangle should be given equal to one side of the other triangle.

3. In Geometry there are two modes of Demonstration, the direct, showing why a thing is so; and the indirect, proving that it must be so,the former being the usual method. Direct Demonstration, as in Prop. 5, is that in which we find intermediate steps which proceed regularly to prove the truth of the proposition: the Indirect Method is only employed, as in Prop. 6, when the predicate of it admits of an alternative, and one of them must be true, because they exhaust every case that can possibly exist. We prove that the alternative cannot be true, and infer, therefore the predicate must be true. With respect to equality between magnitudes, there are two alternatives equal, or unequal; and if we prove that inequality cannot or does not exist, of necessity equality must exist.

USE AND APPLICATION.-Hieronymus the historian, records that THALES of Miletus, who was living 546 B. C., measured the height of the pyramids of Egypt, by observing the shadows which they cast when the shadows were as long as the pyramids were high. This would be the case when the altitude

of a pyramid, or of any other object, and the perpendicular length of the shadow, were equal.

The height of an object and the length of its shadow are the same, when the light, S, which the object, AB, intercepts, is at an elevation of 45°: this condition being observed, the shadow, BC, is equal to the height, BA; because the angles BCA, BAC, being each half a rt. angle, the sides which subtend them are equal. Thus by measuring the shadow CB, we obtain the height BA.

The height would also be obtained by making an observation with a quadrant of altitude; thus,-walk away in a line perpendicular to the altitude of the object, until, at the station C, the quadrant shows A to have an

elevation of 45°: the distance gone over from B to C, or BC, will equal BA, the altitude.

PROP. 7.-THEOR.

Upon the same base and upon the same side of it there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those equal which are terminated in the other extremity.

CONST.--Pst. 1. A st. line may be drawn from one point to another. Pst. 2. A st. line may be produced to any length in a st. line. DEMONSTRATION.-P. 5. The angles at the base of an isosceles triangle are equal, and if the equal sides be produced, the angles upon the other side of the base are equal.

EXP.

SUP.

Ax. 9. The whole is greater than its part.

CASE I.-Let the vertices C and D be without each other.

Join C and D.

ACAD, LACD

LADC:

But ACD > BCD,

ADC / BCD:

and much more is

BDC > BCD.

=

Again, BC BD,

BDC=BCD;

D

B

BDC is both > and BCD, which

CASE II.-Let the vertex D of AADB, be within the ▲ ACB.

CASE III.-When the vertex D is on the side
BC, no demonstration is required.

7 Recap.

Therefore, upon the same
base, &c.

A

Q.E.D.

B

SCHOLIUM.-The argument made use of in this proposition, is the Dilemma, or double Antecedent, in which the truth of the one is impossible, if we admit the truth of the other. The argument called the dilemma may, however, have more than two antecedents; and Whately, p. 72, defines the

true dilemma as "a conditional Syllogism, with several antecedents in the major, and a disjunctive in the minor."

USE. The only purpose for which this proposition is employed, is to prove Prop. 8.

PROP. 8.-THEOR.—(Important.)

If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one, shall be equal to the angle contained by the two sides equal to them of the other.

DEMONSTRATION.-Pr. 7. On the same side of the same base there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity.

Ax. 8. Magnitudes which coincide are equal to one another.

D

G

B

Exp.

1| Hyp.

2 Concl.

DEM. 1 Superp.

2 Hyp.

3 D.2, Hyp.

4 Supp. 1.

2.

E

F

Let the As ABC, DEF, have AB
AC DF, and also BC=EF;
BAC shall equal EDF.

then

DE,

Apply A ABC to ▲ DEF, B on E, and BC
on EF.

:: BC=EF, .. C coincides with F.
Wherefore, BC coincides with EF, BA
and CA shall coincide with ED, FD.
For, suppose that BC coincides with EF,
but BA and CA not with ED, FD, but with
other lines EG, FG,

DEM. 5 Concl.

6 P. 7.

7 D. 2, 3.

E

F

then on this supp., in As EDF, EGF, on the same side of EF, ED shall = E G, and also FD=FG;

which is impossible.

.. since BC coincides with E F, the sides BA, CA, coincide with ED, FD:

Wherefore, BAC must coincide with

EDF;

8 D. 7.

9 Ax. 8. 10 Recap.

Therefore, if two triangles have two sides, &c.

Q.E.D.

SCHOLIUM.-The Equality established is that of the angles,-but the sides being equal, the triangles also must be equal. This is the second criterion of the equality of triangles.

USE.-1. By the aid of this proposition, and of Prop. 22, the angle at a given point C, made by lines from two objects, as A and B, may be determined without a theodolite.

Measure the distances A B,

BC, CA,-and from a scale of
equal parts construct a trian-
gle DEF, the sides of which, A
DE, EF, and FD, will be re-
presentative of the distances
AB, BC, and CA: then with
the semicircle find the number
of degrees in angle F; and as
the triangles ACB, DEF are
similar, that number of de-
grees will also be the measure
of angle C.

2. When the instruments for angular magnitude cannot be employed, by reason of the inequalities of surface, or the difficulty of placing the instruments, this proposition is useful for measuring and cutting angles in a solid body, as in a block of stone, or for bevelling, i. e., for giving the desired