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GEOMETRY.

BOOK I.

§ 1. Definitions—§ 2. First Theorems §3. Parallels-§ 4. Parallelograms 5. Rectangles under the parts of divided Lines-§ 6. Relations of the Sides of Triangles—§ 7. Problems.

SECTION 1. Definitions.

GEOMETRY is the science of extension. The subjects which it considers are extent of distance, extent of surface, and extent of capacity or solid content.

The name Geometry is derived from two Greek words, signifying land and to measure. Hence it would appear that the measurement of land was the most important (perhaps the only) use to which this science was, in the first instance, applied. Egypt is described to have been its birth-place, where the annual inundations of the Nile rendered it of peculiar value to the inhabitants as a means of ascertaining their effaced boundaries. From the Egyptians the ancient Greeks derived their acquaintance with it; and, in the hands of this acute people, it was carried, from a state of comparative nothingness, to a degree of perfection which has scarcely been advanced by succeeding ages. If, however, as a science, Geometry has made but little progress, since it was so successfully cultivated by the Greeks, its uses have been both multiplied and extended. In the present day it embraces the measurement equally of the earth and of the heavens: it forms with arithmetic the basis of all accurate conclusions in the mixed sciences: and there is scarcely any mechanical art, our views of which may not be improved by an acquaintance with it.

The truths of Geometry are founded upon definitions, each furnishing at once an exact notion of the thing defined, and the groundwork of all conclusions relating to it. The leading definitions are as follows:

1. A solid is a magnitude having three dimensions-length, breadth, and thick

ness.

2. A surface is the boundary of a solid, having length and breadth only.

3. A line is the boundary of a surface, having length only.

4. A point is the extremity of a line, ther length, nor breadth, nor thickness. having no dimensions of any kind—nei

5. (Euc. i. def. 4.)* A right line, or straight line, is that which lies evenly between its extreme points.

When the word "line" is used by itself in the following pages, a straight line is

to be understood.

6. Any line of which no part is a right line is called a curve.

If a curve be cut by a straight line in two points, the curve is said to be concave towards that side upon which the straight line lies, and towards the other side, convex.

7. (Euc. i. def. 7.) A plane surface, or plane, is that, in which any two points whatsoever being taken, the straight line between them lies wholly in that surface. 8. A surface, of which no part is plane, is said to be curved.

9. If there be two straight lines in the same plane, which meet one another in a point, they are said to form at that point a plane rectilineal angle.

B

The magnitude of an angle does not depend upon the length of its legs, that is, of the straight lines by which it is contained, but upon the opening between them, or the extent to which they are separated the one from the other. Thus, the angle B A C is greater than the angle BA D, by the angle D A C.

If there be only one angle at the point A, it may be denoted by the letter A alone, as "the angle A;" but if there be more angles at the same point, it becomes necessary to indicate the containing sides of each, in order to distinguish it from the

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others: thus, the angle B A C is distinguished from BAD and DAC. this case the middle letter, as A, always denotes the angular point.

10. (Euc. i. def. 10.) When a straight line standing upon another straight line makes the adjacent angles equal to one another, each of them is called a right angle, and the straight line which stands upon the other is called a perpendicular to it. 11. If an angle be not right, it is called oblique. An oblique angle is said to be acute or obtuse, according as it is less

E

D

C

or greater than a right angle. In the adjoined figure, ABC is a right angle, D B C an acute angle, and EBC an obtuse angle.

12. (Euc. i. def. 35.) If there be two straight lines in the same plane, which,

being produced ever so far both ways, do not meet, these straight lines are called parallels.

13. A plane figure is any portion of a plane surface which is included by a line or lines.

The whole circuit of any figure, that is, the extent of the line or lines by which it is included, is called its perimeter.

14. A plane rectilineal figure is any portion of a plane surface, which is included by right lines. These right lines are called the sides of the figure, and it is said to be trilateral, or quadrilateral, or multilateral, according as it has three, or four, or a greater number of sides.

A trilateral figure is more commonly called a triangle, and a multilateral figure a polygon.

It is further to be understood of rectilineal figures in the present treatise, that the several an

gles are contained towards the interior of the figure; that is, that they have no such angle as the re-entering angle A in the figure which is adjoined. In other words, their perimeters are supposed to be convex externally.

15. A triangle is said to be rightangled, when it has a right angle. Of triangles which are not right-angled, and which are therefore said to be oblique-angled,-an obtuse-angled triangle is that which has an obtuse angle; and an acute-angled triangle is that which has three acute angles.

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also said to be equilateral, when its three sides are equal to one another; isosceles, when only two of its sides are equal; and scalene, when it has three unequal sides. D is an equilateral triangle, E an isosceles triangle, and F a scalene triangle.

A A

The three sides of any the same triangle are frequently distinguished by giving to one of them the name of base, in which case the other two are called the two sides, and the angular point opposite to the base is called the vertex or summit. In an isosceles triangle, considered as such, the vertex is the angular point between the two equal sides, and the base the side opposite to it.

In a right-angled triangle, the side which is opposite to the right angle is called the hypotenuse; and of the other two sides, one is frequently termed the base, and the other the perpendicular.

17. Of quadrilaterals, a parallelogram is that which has its opposite sides parallel, as ABCD. A quadrilateral which has only two of its sides parallel is called a trapezoid, as ABED.

A parallelogram, or indeed any quadrilateral figure, is sometimes cited by two letters only placed at opposite angles: as "the parallelogram A C", "the trapezoid A E." This plan is never adopted, however, where confusion might ensue from it when used, it must always be in such a way as to avoid uncertainty; thus, by "the quadrilateral B D" in the adjoined figure, either ABCD or ABED might be intended, whereas "the quadrilateral A C" is distinct from quadrilateral A E."

"the

18. A rhombus is a parallelogram which has two adjoining sides equal.

C

д

19. A rectangle is a parallelogram which has a right angle. A rectangle

is said to be contained by any two of its adjoining sides; as A C, which is called the rectangle under A B, BC, or the rectangle AB, B C.

20. A square is a rectangle which has two adjoining sides equal. The square described upon any straight line AB, or the square of which A B is a side, is called the square of A B, or A B square, 21. The altitude of a parallelogram or triangle, is a perpendicular drawn to the base from the side or angle opposite. 22. The diagonals of a quadrilateral are the straight lines which join its opposite angles.

23. If through a point, E, in the diagonal of a parallelogram, ABCD, straight lines be

B

A

E

D

drawn parallel to two adjacent sides, the whole parallelogram will be divided into four quadrilaterals; of which two, having the parts of the diagonal for their diagonals, are for that reason said to be about the diagonal; and the two others, A E, E C, are called complements, because, together with the portions about the diagonal, they complete the whole parallelogram ABCD.

is

B

24. A circle is a plane figure contained by one line, which is called the circumference, and such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. This point is called the centre of the circle; and the distance from the centre to the circumference is called the radius, or, sometimes, the semidiameter, because it is the half of a straight line passing through the centre and terminated both ways by the circumference, which straight line is called a diameter. The point C is the centre of the circle ABD; AB is a diameter; and AC a radius or semidiameter.

The truths and questions of Geometry are, for the sake of perspicuity, stated and considered in small separate discourses called Propositions; it being proposed in them either to demonstrate something which is asserted, a proposition of which kind is called a theorem, or to show the manner of doing something which is required to be done, a proposition of which kind is called a problem.

A proposition has commonly the following parts:

3

1o. the enunciation, declaring what is to be proved or done;

20. the construction, inserting the lines necessary thereto;

3°. the demonstration, or course of reasoning; -And,

4°. the conclusion, asserting that the thing required has been proved or done. A corollary to any proposition is a statement of some truth, which is an obvious consequence of the proposition.

A scholium is a remark or observation. The object of a problem, as above stated, is evidently distinct from that of a theorem. If a problem be regarded, however, as demonstrating merely the existence of the points and lines required in its enunciation, it becomes, for our purposes, a theorem certifying the existence of such. And hence has arisen the introduction of problems into the theory of Geometry; for, the existence of the lines and points specified in the constructions of some theorems not being altogether self-evident, it became necessary, either to introduce distinct problems for the finding of such, or to point out the certainty of their existence by the way of theorem and corollary, as occasion offered.

The former plan, exemplified in Euclid's Elements, has been followed by the greater number of geometrical writers; although the problems introduced have not, in all cases, been limited to the very few which are necessary to support the theory. To avoid thus sacrificing unity of purpose, and at the same time not to be wanting to the ends of practical geometry, the problems in the present treatise have been altogether separated from the theorems; and the requisite support has been supplied to the latter, in the second of the two ways above mentioned.

The existence of the following lines, &c. will be taken for granted; and they will, therefore, be referred to by the name of POSTULATES.*

passes through two given points, A, B.
1. A straight line, which joins or

B

2. A circle, which is described from a given centre, C, with a given radius, CA.

Things required; from the Latin postulo, to require.

3. A point which bi

sects a given finite straight line, A B,

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of it; the two together shall be double of the third magnitude.

10. Straight lines which pass through

that is, which divides it into two equal the same two points lie in one and the

parts.

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1. Things, which are equal to the same, are equal to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be taken from equals, the remainders are equal.

4. The doubles of equals are equal. 5. The halves of equals are equal. 6. The greater of two magnitudes, increased or diminished by any magnitude, is greater than the less increased or diminished by the same magnitude.

7. The double of the greater is greater than the double of the less.

8. The half of the greater is greater

than the half of the less.

9. If there be two magnitudes, and a third, and if one of them exceed the third by as much as the other falls short

Authorities, or things having authority; from a Greek word.

same straight line.

11. Magnitudes, which may be made to coincide with one another, that is, to fill exactly the same space, are equal to one another.

The converse of this last axiom is likewise true of some magnitudes. In what follows, it will be assumed, with regard to straight lines and angles; i. e. it will be assumed that if two straight lines are equal, they may be made to coincide with one another, and the same of two angles.

SECTION 2. First Theorems.

PROP. 1. (EUc. i. Ax. 11.) All right angles are equal to one another.

Let the angles ABC, DEF be each of them a right angle; the angle A B C shall be equal to the angle DEF.

I

B

Produce C B to any point G, and FE to any point H. Then, because ABC is a right angle, it is equal to the adjacent angle A BG (def. 10.); and because DE F is a right angle, it is equal to DEH.

From E draw any straight line E K. Then, because the angle KEH is greater than DE H, and that DE H is equal to DEF, KEH is greater than DEF: but D E F is greater than K E F: much more, then, is K E H greater than KE F.

Now, let the angle ABC be applied to the angle DEF, so that the point B may be upon E, and the straight line B C upon E F; then (ax. 10.) BG will coincide with E H. And, B G coinciding with EH, BA must also coincide with ED; for, should it fall otherwise, as E K, the angle ABG would be greater than ABC, by what has been already demon strated, whereas, they are equal to one

another.

Therefore, B A coincides with ED, and the angle A B C coincides with the angle DE F, and (ax. 11.) is equal to it.

Therefore, all right angles are equal to one another, which was to be demonstrated.

PROP. 2. (EUc. i. 13 & 14.)

The adjacent angles, which one straight line makes with another upon

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are either two right angles, or are together equal to two right angles. For, if they are equal, then is each of them (def. 10.) a right angle.

But, if not, from the point B draw BE perpendicular to CD (Post. 5.). And because the angle EBD is equal to the two angles, EBA, ABD, to each of these equals add the angle EBC: therefore, (ax. 2.) the two angles EBC, EBD are equal to the three angles EBC, EBA, ABD. And in the same manner it may be shewn, that the two angles A B C, ABD, are equal to the same three angles. Therefore, (ax. 1.) the angles A B C, ABD, are together equal to the angles E B C, E B D, that is, to two right angles.

Next, let the straight line A B make with the two straight lines, B C, B D, at the same point B, the adjacent angles ABC, ABD together equal to two right angles: BC, BD shall be in one and the same straight line.

For, let B F be in the same straight line with BC: then, by the first part of the proposition, because A B makes angles with C F upon one side of it, these angles, viz. ABC, A BF, are together equal to two right angles. But ABC, A B D are also equal to two right angles; therefore, (ax. 1.) A BC, A BD together are equal to ABC, ABF together; and, ABC being taken from each of these equals, the angle A B D is equal to ABF (ax. 3.) Therefore B D coincides with BF; that is, it is in the same straight line with B C.

Therefore, &c.*

Cor. 1. If from a point in a straight line there be drawn any number of straight lines upon one side of it, all the angles (made by each with that next to it) shall be together equal to two right angles.

Hence the adjacent angle ABD is sometimes said to be supplementary to ABC; one angle being called the supplement of another, when together with that ether it is equal to two right angles.

Cor. 2. Any angle of a triangle is less than two right angles.

PROP. 3. (EUC. i. 15.)

If two straight lines cut one another, the vertical or opposite angles shall be equal.

Let the two straight lines A B, CD, cut one another in the point E: the vertical D angles AED, BEC, as also the vertical

angles AEC, BED, shall be equal to one another.

Because the angles AEC, AED are adjacent angles made by the straight line AE with CD, they are (2.) together equal to two right angles; and for the like reason, the angles AEC, CEB, are together equal to two right angles; therefore, (ax. 1.) the angles AEC, AED together are equal to the angles AEC, CEB together. From each of these equals take the angle A E C, and the angle A E D is equal to the angle CE B. (ax. 3.) In the same manner it may be shown that the angles AEC, BE D are equal to one another. Therefore, &c.

Cor. (Euc. i. 15. Cor. 2.) If any number of straight lines pass through the same point, all the angles about that point, (made by each with that next to it,) shall be together equal to four right angles.

PROP. 4. (EUc. i. 4.)

If two triangles have two sides of the one equal to two sides of the other, each to each, and likewise the included angles equal; their other angles shall be equal, each to each, viz. those to which the equal sides are opposite, and the base, or third side, of the one shall be equal to the base, or third side, of the other.

Let ABC, DEF be two triangles, which have the two sides A B, A C, equal to the two sides D E, DF, each to each, viz. A B to D E, and A C to D F, and let them likewise have the angle BAC equal to the angle ED F: their other angles shall be equal, each to each, viz. ABC to DEF, and AC B to DFE, and the base BC shall be equal to the base EF.

For if the triangle B ABC be applied to the triangle DE F, so that the point A may be upon D, and the

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