8. 61. Important.-If two As have each the three sides eq. the Z contained by two equal sides in one ▲ equals the contained by the two corresponding eq. sides in the and the ▲s are equal.

other A,

SCH.-The second criterion for equality of As.

USE 1. To determine

without a theodolite the at a given., made by Lines from two objects. 2. To measure and cut angles in a solid body.

13. 71. The s made by one st. L. with another on one side of it, are either rt. s, or together = two rt. S.

SCH.-Any number of Lines converging to a, in a L on one side of it make thes together = two rt. /s. Supplement and complement of an explained. USE Pr. 13 of frequent Use in Trigonometry, to determine the third, when two s are given.

14 73. Conversely.—If at a. in a st. L., two lines on the opp. sides of it, make the adj. /s together = 2 rt. s, the two

Lines form one and the same st. L.

15. 74. If two st. Lines cut one another, the opp. or vert. <s shall be equal; and conversely.

COR. 1. The s formed by two lines crossing each other are together = 4 rt. /s.

2.

All the /s formed by any number of lines diverging from a com. centre are together = 4 rt. s.

SCH.-A developement of the def. of an . The Converse true.

USE 1.—To find the distance between two inaccessible objects; 2. To make one elastic ball strike another by reflection; and 3. to determine the number and kind of polygons which on being joined cover a given space. 16. 76. If one side of a ▲ be produced, the ext. < is > either of the int. opp. s.

SCH. Each of a ▲ is < the supplement of either of the others. USE 1 Among other conclusions,-only one perp. from a, to a given L. 2. Prop 16, of great use in reducing As and other rectil. figures to rectangles.

17. 78. Any two s of a ▲ are together < 2 rt. ≤ s.

Explanatory of Ax. 12. Both Pr. 16 & 17 included in Prop. 32.
The gr. side of every ▲ is opp. to the gr. Z.

18. 79.

An instance of the argument " à fortiori."

19. 79. Conversely.-The gr. side of every ▲ is subtended by the

gr. 4.

of a ▲ is, >, or

the other side oppo

SCH.-Prop. 5, 6, 18 & 19, combined, prove, "One <another, as the side opposed is >,, or sed; and "vice versa." USE 1. The Perp. is the shortest L from a. to a given L. 2. From one, only two eq. lines to a given L. can be drawn. 3. All heavy bodies free to move seek the. nearest the earth's centre. 4. To construct a ▲, having the base, the less at the base, and the diff. of the sides given.

20. 81. Any two sides of a ▲

third side.

are together greater than the

COR. The diff. of any two sides of a ▲ is less than the remaining side. N.B.-More assumed in the Cor. than is expressed in Ax. 5.

=

2.

USE 1.-Of all lines from one to another and reflected to a third; those the shortest which make the of incidence = the of reflection. Natural causes act by the shortest lines; hence, by means of a mirror to construct a ▲ of which the Perp. is representative of the height of an object.

21. 84. If from the ends of a side of a A two Ls be drawn to a. within the A. these lines are the other two sides, but contain a greater ▲,

Applied in Optics, Astronomy, and Architecture.

24. 91. If two As have two sides of one two sides of the other, but the contained by the two sides of the one > the contained by the two sides of the other, the base of that which has the gr. shall be > the base of the other.

one

= two base of one

25. 92. Conversely. If two As have two sides of sides of the other, each to each, but the > the base of the other, the opp. the gr. base shall be the opp. the less base.

Pr. 4, 8, 24 and 25, may be combined; "If two As have each two sides two sides, the third side of the one will be >, <, or the third side of the other, as the opposed in one is >, <, or the opposed in the other; and vice versâ.

26. 93. Important.—If two ▲s have two sand a side of the one = two s and a side of the other; the other sides the third of the one

shall be eq. each to each, and

the third of the other.

SCH.-The third criterion of the equality of As. In two or more As, any three parts of which one must be a side, being given equal, the equality of the other parts will follow.

USE 1.-Applied to measure inaccessible distances ;-2 & 3, by the Theory of Representative Values to find the distance of two stations ;-4, to construct an isosc. A. the vert. and perp. height of the ▲ being given.

A L., perp. to one parallel, is also perp. to the

27. 97. If a L falling on two other lines makes the alternate s equal, these two lines are parallel.

SCH.-Since some curved lines, though they never intersect, are not parallels, another demonstration is given.

28. 99. If a L falling on two other lines makes the ext.

=

the int. and opp. ▲ on the same side of the line; or the int.s together on the same side = 2 rt. ≤s; the two lines shall be parallel.

SCH.-The principle in Ax. 12 really is,-that two st. lines intersecting cannot both be || to the same L.

29. 100. If a L fall on two || st. lines, it makes the alternate

equal; and the ext. = the int. and opp. ▲ on the same side; and the two int. s on the same side together = two rt. Ls.

Converse of Pr. 27 & 28.

SCH.-Methods of expressing Ax. 12, Definition of Parallel Lines. USE.-Pr. 27, 28 and 29 are applied to determine the earth's circumference. 30. 102. Lines || the same L are parallel to each other.

COR.-Two lines || the same L cannot pass through the same point; equivalent to Ax. 12.

32. 105. Very important.-If a side of a ▲ be produced, the ext. L = the two int. & opp. Zs; and the three int. Zs 2 rt. s.

of every together

COR. 1.-All the ints of any rectil. fig., + 4 rt. s twice as many rt. Zs as the fig. has sides. This Cor. is of uuiversal extent.

2. All the ext. /s of any rectil. fig. are together = 4rt. s, Applicable only to convex figures; not to figures with re-entrants. 3.-If two As have two s of the one two s of the other; the third of the one the third of the other.

SCH.-Lardner's Euclid gives twenty four corollaries.

USE. This Theorem employed, 1. To determine the Parallax of a heavenly body; 2, To give the representative height of a mountain; and 3, to construct any regular right-lined figure.

33. 109. The lines joining the extremities of eq. and parallel lines, towards the same parts, are also

eq.

and parallel.

USE. To ascertaiu the perp. height of a mountain, as well as the distance from the base to the foot of the perp.

34. 110. The opp. sides and sof

7s are eq. to one another, and the diagonal bisects them; and conversely.

SCH.-If a quadril. fig. have any two of certain ten data, it will also have the others. By combining the ten, 360 questions are raised. USE. 1.-The construction of the parallel ruler depends on this Prop. It is also useful, 2, to divide a line into any number of eq. parts; 3, to construct the Sliding Scale, called the Vernier or Nonius for measuring minute parts; 4, to obtain the distance between two objects; 5, to continue a st. line when an obstacle intervenes; 5, to divide a eq. pts. from a in one of the sides, &c.

into two

35. 113. Parallelograms on the same base and between the same parallels are equal, or rather equivalent, to one another.

SCH.-The equality of s proved by the Method of Indivisibles.
USE.-The Prop. applied to convert a into a rect. of eq. area.

The

linear units in the base multiplied by the linear units in the altitude of a gives the Area.

36. 116. Parallelograms on eq. bases, and between the same s are

equal.

USE. The Construction of the Diagonal Scale, and its application.

37. 118. Triangles on the same base and between the same ||s, are

equal.

Half the product of the base and altitude of a ▲ gives the Arca.

38. 119. Triangles on eq. bases and between the same ||s, are eq. to one another.

SCH.-By dividing the base into eq. pts., and joining the .s of division to the vertex a A is divided into eq. parts.

USE.-This Prop. also enables us from any. in a side of a A to divide it into two eq. parts.

39. 120. Eq. As on the same base and on the same side of it are between the same parallels.

The loci of the vertices of eq. As on the same base, form a st.line 40. 121. Eq. As on eq. bases in the same st. line, and towards the same parts, are between the same parallels.

SCH. COR. 1.-A || to the base of a ▲ through the middle. of one side will bisect the other side.

2.

The lines joining the middle. s of the three sides divide the

A into four eq, As.

3.—The L joining the points of bisection of each pair of sides
is eq. to half the third side.

4.-A trapezium : = a

of the same alt. and of which the

base is half the sum of the || sides.

The Area of a trapezium the sum of the || sides X the altitude.
The Area of a Square the Square of the lineal units in one side.
Many other corollaries may be derived from Prop. 40.

41. 123. Important.—If a ☐ and a ▲ be on the same base and
between the same ||s, the shall be double of the

A; and conversely.

USE.-The Area of any figure iresolvable into ▲s depends on this Propo-
sition; which enables us to find,-1, the area of a ▲; 2, the area
of any rectilineal figure; 3, the area of a polygon; and 4th, the area
even of a circle.

43. 126. The complements of the

7s which are about the diam.

of any, are eq. to one another.

COR.-The 7s about the diag. and their complements are equiangular
with the whole.

USE. To find a = a given

and having one side a given line.

47. 135. Most important.-In any rt. d▲, the square on the
side opp. to the rt. is eq. to the squares on the sides
containing the rt. ▲.

COR. 1.-Hence, if the sides of a rt. d▲ be given in numbers the
hypotenuse may be found.

= their

2.-If the hyp. and one side be given, the other side may be found.
3. If any number of squares be given, a square may be found
sum; or the multiple of a sq. may be ascertained; or the difference
of two squares; or a sq. may be made the, &c. of a given sq.
4.-If a perp. be drawn from the vert. of a ▲ to the base, the difference
of the squares of the sides the difference of the squares of the
segments.

5.-If a perp. be drawn from the vertex to the base or base produced, the
sums of the squares of the sides and alternate segments are equal.

138. SCH.-A Practical Illustration of Prop. 47. I.

139. USE 1.-Combined with other propositions, the 47. I. is applied 1o,
to make a rectil. fig. similar to a given rectil. fig. ; 2°. to make
a double, or the half of another; 2. To construct the Chords,
Natural Lines, Tangents and Secants of Trigonometrical Tables;
3. To find right triangular numbers; & 4, 5, To ascertain
heights and distances from the curvature of the earth.

48. 142. Conversely.—If the sq. on one of the sides of a ▲ be
eq. to the squares on the other two sides of it, the
contained by the two sides is a rt. .