of the summits of Human Wisdom, and as he looks around, either to survey the Country already traversed, or to take note of the Heights still rising before him, he may inscribe upon his work, in the spirit of devotion;

Laus Deo,

I thank God, and take courage.

SO NOW SO EVER.

EUCLID'S PLANE GEOMETRY PRACTICALLY APPLIED.

Section.

Page.

Definitions. 1-35
Definition.

Postulates.

Axioms. I. 1-7

Synoptical INDEX to Books I-VI.

Part I. containing Bks. I.—II.

INTRODUCTION.

Gradual Growth of Geometry and of the Elements of Euclid.
Symbolical Notation and Abbreviations.

Explanation of some Geometrical Terms.

Nature of Geometrical Reasoning.

Application of Algebra and Arithmetic to Geometry.
On Incommensurable Quantities.

Written and Oral Examinations, and Plan of Examination.

BOOK I.

The Geometry of Plane Triangles.

pp. 37-43

43-44

1-3

44

44-45

45-46

45-46

II. 8-12

EUCLID added three other Postulates.

Applicable to number and quantity as well as to magnitude.

Applicable especially to magnitude.

Super-position. Illustration of Lines parallel

and not parallel. Lines converging and diverging. Angles interior, exterior, opposite, adjacent, vertical and alternate.

FOURTEEN PROBLEMS :

Prop. 1, 2, 3;--9, 10, 11, 12 ;-22, 23 ;-31 ;-42;—44, 45, 46. Prop 1. p. 47. To describe an equil. A on a given st. Line. USE OR APP. 1°,-To solve 2, 3, 9, 10 & 11, I; 2°. draw an isosc. A.; 3°. approximate to an oval; and 4° to measure an inaccessible distance. 2. 48. To draw from a. a st. Line a given st. Line.

SCH.-Eight Solutions of this Problem.

3. 50. From the gr. of two Lines to cut off a part the less.

SCH.-To lengthen the less to eq. the greater; USE.—to construct a Scale of eq. parts, and to apply the principle of Representative values.

9. 63. To bisect a rectil. ▲, i. e. to divide it into two eq. parts. SCH.-To bisect an arc of a; & by successive bisections to divide an into any parts indicated by a power of 2. USE. To bisect the base of an isosc A; and to construct the Mariner's Compass.

10. 66. To bisect a give st. Line.

SCH-By successive bisections to divide a L. into parts, indicated by a power

of 2.

11. 67.

From a . in a L. to draw a L. at rt. ▲s to it.

COR. Two lines cannot have a common segment.

SCH.-To draw a Perp. from the extremity of a L. USE 1.-To construct a square; 2. On a given Line to describe an isosc. ▲ of which the perp. height: = the base.

12. 69. From a . without a L. to draw a Perpendicular.

SCH.-When the. is over the extremity of the L. USE. This Prop. indispensable to all artificers, &c.

22. 86. To make a ▲ of which the sides shall be eq. to three given st. Ls., any two being > than the third.

SCH.-Assumed that two Os will have at least one. of intersection. USE. Of most extensive use,-to make one rect. fig. = or similar to another; And, on a given L. to describe an isosc. ▲ with sides each twice the base; &c.

USE 1.

23. 88. At a given. to make a rectil. = a given rectil. 2. Of the widest use in Practical Mathematics; 2. To construct a Line of Chords; and by it to make an of a certain magnitude; At the end of a L. to draw a Perp.; to find the measure of an ; and to draw As with certain parts given.

31. 103. Through a . to draw a st. L. parallel to a given st. L.

SCH.-Demonstration of the 12th Axiom. USE.-Prop. 31 required in all branches of Practical Mathematics;-it enables the Surveyor to ascertain inaccessible distances.

42. 125. To descr. a = a given ▲, and having one ▲ = a given Z.

SCH.-Or, a ▲ a given, and having an = a given 44. 127. To a given L. to apply a

an a given Z.

USE.-Geometrical Division illustrated.

.

= a given ▲, and having

45, 130. To descr. a

= a given.

= a given rectil. fig., and having an

USE. To measure the superficial content of any rectil. fig. ; 2, To change any rectil. fig. into a A, and then into a rect. of eq. Area; and 3. To straighten a crooked boundary without changing the dimension.

46. 132. To describe a Square on a given st. L.

COR. 1. The squares on eq. Lines are eq.; 2. Every parallelogram with one rt.

and conversely.
has all its s rt s.

SCH.-Given the diagonal to construct a square.

USE.-The Geometrical Square; its construction; its use in ascertaining inaccessible distances, as heights.

THIRTY FOUR THEOREMS.-ONE LEMMA ;

Prop. 4, 5, 6, 7, 8;-13, 14, 15, 16, 17, 18, 19, 20, 21 ;-24, 25, 26;-Lemma ;27, 28, 29, 30,-32, 33, 34, 35, 36, 37, 38, 39, 40, 41,; —43; 47, 48. 4. 52. Important.—If two As have each two sides and their included eq. the bases and others are eq. and the As equal.

SCH.-The equality is perfect, not in Area only. USE.--The first criterion for establishing the equality of As; very frequently applied; and useful, along with the Theory of Representative Values, for ascertaining inaccessible distances.

5. 54. The s at the base of an isosc▲ are eq.; and if the eq. sides are produced, the s on the other side also are equal.

COR. Every equal triangle is also equiangular.

6. 57. Conversely. If two /s of a ▲ are eq. the sides opp. the eq. s also are equal.

COR. Every equiangular ▲ shall be equilateral.

SCH. 1, 2.-Converse Theorems not universally true. 3. Two modes of Demonstration, direct and indirect. USE. To determine the Height of an object by its shadow.

7. 59.

On the same side of the same base there cannot be two As, with sides terminated in one extremity equal, and also the sides equal terminated in the other extremity. The Dilemma, or Double Antecedent, Prop. 7, used only to prove 8, I.