but the law of each series must be known, in order to bring those relations into view. The relations of geometry, however, are presented to the eye;-they are realities; the business of the world in miniature: it reasons from certain data, and furnishes the best model of reasoning in things less certain.

It was not design that brought arithmetic to occupy the exclusive ground it has held in the schools. Children could always repeat several terms of the natural series of numbers, and merchants could always sum and sever their gains and losses: in addition to this, the Moorish system of notation had come into use, carrying the series beyond any assignable limit. At the time of the revival of letters, a basis was thus provided for the study of arithmetic; and to read, write, and cipher, was esteemed accomplished scholarship. No regard was had to the mechanical operations of the world, or to the laws of the universe. Since that time, arithmetic has been compiled and re-modelled into a thousand forms, and algebra superadded; but geometry, which should be, at least, co-ordinate with arithmetic, has been neglected, or left to the speculative scholar. Yet these elements belong to the operatives-to the employments of men; they arm men with the skill and force of Nature, and imbue them with her wise designs; they verify the adage that "knowledge is power."

There is a solemn voice in the natural truths of geometry, which calls upon School Officers, Professors, and Teachers, to sow the seed, to scatter it broadcast over this national husbandry, to sacrifice the distinctions of learning to the perpetuation of a wise popular government, through the medium of an efficient elementary education. And this voice would be obeyed, and this motive deemed sufficient, if something great were required to be done; but when it is merely to put into the hands of children the text of Euclid, to be read and recited, will they not say, "Where is the use of it?" Where then is the use of scattering so much good wheat over the fields? Is it all lost? No! the stoutest doubter expects twenty to one of the same kind. By the same rule, therefore, we may have twenty Euclids to one from the Common Schools and Academies.

New York, March, 1846.

SECOND LESSONS IN GEOMETRY.

BOOK I.

Definitions.

1. GEOMETRY is the science which treats of the similarity, equality, difference and proportions of magnitudes and of figures of extension.

2. A point is a position, or station in a line, at the extremities of a finite line, also at the meeting and intersection of lines: but it is not the measuring unit, nor any part of the measure of a line.

Cor. Hence, points are by position, central, angular, sectional, or ex

treme.

3. A finite line is that of which the extreme points are given.

Note. There are two classes of lines; namely, straight and curved: of curves there are several species; but the circle alone will be here considered. Lines have lengths, but no other dimensions.

4. A straight line is the path of a point, without curve or anglo.

Cor. Two straight lines cannot meet and part and meet again: they cannot have a common segment; but, meeting in two points, or coinciding in part, they shall coincide in all their length.

Note. Straight lines have certain relations to one another from their position; namely, perpendicular, meeting, insisting, parallel, and intersecting. One line is perpendicular to another when it makes the adjacent angles equal, or when it pends, or hangs upon the other as the plumbline upon the level: lines meet when they touch and do not cut one another; one line insists upon another when it stands upon a point in the other: one line is parallel to another when it is in the same plane with the other, and at all points equidistant from it: lines intersect one another when they pass through the same point.

5. A circular line is the path of a moving point about a stationary one, at the same uniform distance from it.

6. A superficies is the upper or outside face-the surface: it has two dimensions-length and breadth: it is bounded by a line, or lines; and the intersection of two superficies is a line.

7. A plane is a superficies described by the lateral motion of a straight line; or, by its rotary motion about one of its extreme points.

8. A plane angle is the rotary declination of one straight line from another, about a stationary point in which they meet.

NOTE. Angles are, by position, adjacent or opposite, interior or exterior, vertical or alternate; by magnitude, they are acute, right, or obtuse; the acute and obtuse are called oblique angles. Salient and re

entrant angles are the outward angles in fortifications; the former are greater and the latter less than half the compass of the angular point. 9. An acute angle is any declination of two straight lines, smaller than one-fourth of the compass of the angular point.

10. A right angle is the declination of two straight lines to one-fourth of the compass of the angular point.

11. An obtuse angle is any declination of two straight lines greater than one-fourth but less than half the compass of the angular point.

Cor. When the declination of two straight lines is equal to half the compass of the point in which they meet, they form no angle, but are in one straight line.

12. A plane figure is any form of a superficies,—it is bounded by at least three straight lines.

13. A circle is a plane figure enclosed by a uniformly curved line, called the circumference.

14. The centre of a circle is a point within it, equidistant from every point of the circumference.

15. A radius is any straight line drawn from the centre to the circumference of a circle: therefore all radii of the same or equal circles are equal to one another.

16. A diameter of a circle is a straight line drawn through the centre to the circumference on either side;—of a parallelogram is that straight line which joins opposite angles.

17. A semicircle is a figure contained by the diameter and half the circumference.

18. An arc is any part of the circumference of a circle,-a chord is the straight line which joins the extremities of the arc.

19. A sector is the figure contained by two radii and an arc.

20. A segment of a circle is a part cut off by a chord,—a segment of a line is a part cut off or distinct.

21. Rectilineal figures are those which are enclosed by straight lines: trilaterals have three sides; quadrilaterals, four, &c.

22. Trilateral figures, or triangles, have six parts; namely, three sides and three angles; from which they take the following names: 23. An equilateral triangle has three equal sides.

24. An isosceles triangle has two equal sides.

25. A scalene triangle has three sides unequal.

26. An acute-angled triangle has its three angles acute.

27. A right-angled triangle has one right angle.

28. An obtuse-angled triangle has one obtuse angle.

29. Quadrilateral figures, or quadrangles, are contained by four straight lines: of this kind are the square and oblong, the rhombus and rhomboid, the trapezoid and trapezium.

30. A square has four equal sides, and four right angles.

31. An oblong, or rectangle, has four right angles, and its opposite sides equal and parallel.

32. A rhombus has four equal sides, of which the opposite are paraland four oblique angles, of which the opposite are equal.

lel;

33. A rhomboid has its opposite sides and angles equal, and all its angles oblique.

34. A trapezoid has two of its opposite sides parallel.

35. All other quadrilateral figures are called trapezia.

36. Multilateral figures, or polygons, are contained by five or more sides; but the triangle and square may be included in the series. They are called regular when they are equilateral.

NOTE. The series of regular polygons are equiangular: it may be said to begin with the triangle, and end with the circle.

The Greeks named the regular polygons from their angles, viz:

A trigon has

three equal angles.

A tetragon has

four

A pentagon has

A nonagon has

A decagon has

An undecagon has eleven "

A dodecagon has twelve "

A trisdecagon has thirteen equal angles; a quindecagon, fifteen; and so upwards to infinity, which may be represented by the circle.

37. The perimeter is the sum of the lines which bound a figure: and it may be remarked, although it will be hereafter proved, that under equal perimeters, regular polygons contain greater areas than figures contain whose sides are unequal.

Also, the perimeters of regular polygons being equal, that perimeter which has the greater number of sides, contains the greater area; and the circle contains the greatest area within equal bounds.

38. A theorem is a proposition which requires to be proved.
39. A problem is a proposition requiring something to be done
40. A corollary is an inference from a previous demonstration.
41. A scholium is a remark explanatory of the preceding subject.

POSTULATES, OR PETITIONS.

The geometer requires permission to move freely in space, viz: 1. To draw a straight line from one point to another.

2. To produce a terminated straight line.

3. To describe a circle about any point with any radius.

AXIOMS, OR MAXIMS.

1. Things which are equal to the same are equal to each other.

2. Add equals to equals, the sums will be equal.

3. Take equals from equals, the remainders will be equal.

4. Add equals to unequals, the sums will be unequal.

5. Take equals from unequals, the remainders will be unequal.

6. Doubles of the same are equal to each other.

7. Halves of the same are equal to each other.

8. Things which coincide, or fill the same space, are equal to each other.

9. The whole is equal to all its parts, and greater than its part.

10. All right angles are equal to one another; and so are all angles measured by equal arcs to equal radii.

11. Two intersecting straight lines cannot both be parallel to the same straight line, or to each other.

Illustration of the Definitions.

The angular point is marked by a letter, and when three or more lines meet in a point, their extremities are lettered; and in naming the angle, the letter at the angular point is read between the other two.

P, after the number of the proposition, stands for problem; Th. for theorem; p., after recite, for proposition; ax., axiom; cor., corollary; def., definition; pos., postulate; hyp., hypothesis. The first figure refers to the proposition; the latter, to the book.