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cor. 5

are, together, equal to four right angles (e) Angles, which have the sides of the one parallel, or perpendicular, or equally inclined to the sides of the other, in the same order, are equal 14

(C) of parallel Straight Lines. Difficulty in the theory

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sch. 11 (a) Straight lines which are perpendicular to the same straight line are parallel; and conversely (b) A parallel to a given straight line may be drawn through any given point without it; but through the same point there cannot be drawn more than one parallel to the same given straight line cor. 11 (c) Straight lines which make equal angles with the same straight line towards the same parts are parallel; and conversely (d) If a straight line falls upon two other straight lines, so as to make the alternate angles equal to one another, or the exterior angle equal to the interior and opposite upon the same side, or the two interior angles upon the same side together equal to two right angles, those two straight lines are parallel cor. 13 (e) And conversely, if a straight line falls upon two parallel straight lines, it makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite upon the same side, and the two interior angles upon the same side together equal to two right angles (f) If a straight line falls upon two other straight lines, so as to make the two interior angles upon the same side together less than two right angles, those two straight lines are not parallel, but may be produced to meet one another upon that side cor. 13

cor. 13

(g) Straight lines, to which the same straight line is parallel, are parallel to one another

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(D) Rectangles under the parts of divided Lines, II. § 5 19, 20 The theorems of which section are briefly expressed as follows:

(a) A × (B + C + D) = A B + AC
+AD.

(b) (A+B) × BAB+ B2.
(c) (A+B)2 = A2 + B2 + 2 A B.
(d) (AB)2 = A2 + B2 2 A B.
(e) (A+B)2 - (A — B)2 = 4 A B.
(ƒ) (A + B)2 + (A — B)2
+B).

(g) (A + B) x (AB)

= 2 (A2

A - B2.

(E) Of Straight Lines which are propor tionals.

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(a) If four straight lines are proportionals, the rectangle under the extremes is equal to the rectangle under the means; and conversely, if two rectangles are equal to one another, their sides are proportionals, the sides of the one being extremes, and the sides of the other means 63 (b) If A, B are two straight lines, and A', B' other two, the rectangle A× A' shall be to the rectangle BX B' in the ratio which is compounded of the ratios of A to B and A' to B' (c) If A, B and A', B' are proportionals, AXA' is to BX B' as A2 to B2 cor. 63 (d) If A, B, C are proportionals, A is to C as A2 to B2 (e) If A, B, C, D, and also A', B', C',

63

cor. 63

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cor. 63

D', are proportionals, the rectangles Ax A, B x B, C x C', and DX D' are also proportionals cor. 63 (f) The squares of proportional straight lines are proportionals; and conversely (9) If A, B are two straight lines, A', B' other two, and A", B" other two, the rectangular parallelopipeds AX A' XA", B'xB" shall be to one another in the ratio which is compounded of the ratios of A to B, A' to B', and A" to B" 143 [(h) If the ratios of A to B, A' to B', and A" to B" are all equal, A x A′ × A′′ is to B x B'x B" as A3 to B3.] [) If A, B, C, D are in continued proportion, A is to D as A to B.] [(k) If A, B, C, D, and also A', B, C, D′, and also A", B", C", D", are proportionals, the rectangular parallelopipeds AX A' x A", B x B′ x B", CX C' × C", D X D' × D", are also proportionals.]

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(F) Of Straight Lines harmonically divided. (a) If A B, AC, AD are harmonical progressionals in the same straight line, and A B the least, DC, DB, ĎA shall likewise be harmonical progres sionals

means

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(b) The same being supposed, and the mean A C being bisected in K, KB, KC, KD are proportionals; and, conversely, if KB, KC, K D are proportionals, and if K A is taken in the opposite direction equal to K C, AB, A C, A D shall be in harmonical progression (c) The same being supposed, DA, DK, DB, D C are proportionals (d) The harmonical mean between two straight lines is a third proportional to the arithmetical and geometrical cor. 69 (e) If any four straight lines pass through the same point, and lie in the same plane, to whichsoever of the four a parallel is drawn, its parts intercepted by the other three shall be to one another in the same ratio (f) If a parallel is drawn to any one of four harmonicals, equal parts of such parallel are intercepted by the other three; and, conversely, if four straight lines pass through the same point, and if a parallel drawn to one of them has equal parts of it intercepted by the other three, the four are harmonicals cor. 70

70

(9) Harmonicals divide every straight line, which is cut by them, harmonically 70 (h) The two sides of a triangle, the straight line which is drawn from the

vertex through the middle point of the base, and the straight line which is drawn through the vertex parallel to the base are harmonicals

cor. 70 (i) The two sides of a triangle, and the lines which bisect the vertical and exterior vertical angles, are harmonicals cor. 71

(G) Problems relating to Straight Lines. (a) Through a given point to draw a straight line

1. Perpendicular to a given straight line; several methods 24, 25 2. Parallel to a given straight line 26 When many parallels are to be

drawn, See II. 49. 3. Which shall pass through a point which is the point of concourse of two given straight lines, but is without the limits of the draught (See "Centrolinead.") 75

4. So that its parts intercepted by two given straight lines shall be to one another in a given ratio 74 5. Which shall form with the parts cut off by it from two given straight lines a triangle equal to a given triangle

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(e) To find two straight lines, there being given the

1. Sum, and difference.

2. Sum of squares, and difference
of squares.

3. Sum, and sum of squares.
4. Difference, and sum of squares.
5. Sum, and difference of squares.
6. Difference, and difference of
squares.

7. Ratio, and rectangle.
8. Sum, and ratio.

9. Difference, and ratio. 10. Sum, and rectangle. 11. Difference, and rectangle. 12. Sum of squares, and ratio. 13. Difference of squares, and ratio. 14. Sum of squares, and rectangle. 15. Difference of squares, and rectangle 123 (f) To draw the shortest distance between two given straight lines which do not lie in the same plane 154 Subcontrary section of an oblique cone 229 of an oblique cylinder

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[Symmetrical. Any two solids are said to be symmetrical, when they can be placed upon the two sides of a plane, so that for every point in the surface of the one, there is a point in the surface of the other, in the same perpendicular to the plane, and at the same distance from it; also, the solids, when so placed, are said to be symmetrically situated, with regard to the plane*. Of such solids it may be demonstrated

1. That if any polyhedron whatever be inscribed in, or circumscribed about the one, a polyhedron symmetrical with it may be inscribed in, or circumscribed about the other.

2. That they are equal to one an other, and have equal surfaces.] Symmetrically divided; a plane (or solid) figure is said to be symmetrically divided by a straight line (or plane), when, for every point in the contour of the figure which is upon one side of that line (or plane), there is another point in the con

* According to a similar description, it is evident that two plane figures may be said to be symmetrical, and symmetrically situated with regard to a straight line which is in the same plane with them; but such figures are also similar, and may be made to coincide, which is not the case with symmetrical solids.

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(d) To circumscribe a sphere about a given tetrahedron

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Regular. See "Regular Polyhedron." Third proportional (see " Straight Line" def.33 Third harmonical progressional def. 68 [Touch; (a) a straight line is said to touch a curve in any point, when it meets, but does not cut the curve, in that point*. Such a line is called a tangent.

(b) One curve is said to touch another
in any point when they have the
same tangent at that point.

(c) A plane is said to touch a curved
surface in any point or line, when it
meets, but does not cut, the surface in
that point or line: such a plane is
called a tangent-plane, and the point or
line in which it touches the surface, is
called the point or line of contact.
(d) One curved surface is said to touch
another in any point or line, when they
have the same tangent plane at that
point or at every point of that line.
(e) The following examples may be
given of the contact of surfaces-

1. A plane touches

The surface of a sphere in a point, the convex surface of a cylinder in a straight line which is parallel to the axis, the convex surface of a cone in a slant side.

2. A spherical surface touches A spherical surface, whether externally or internally, in a point; the convex surface of a cylinder or

Some curves are of a winding or serpentine form, having the curvature or bending now towards the one side, now towards the other. In such curves, there are always one or more points of contrary flexure, i. e. points where the curvature which had before been towards one side changes to the other side. Through such points no straight line can be drawn so as not to cut the curve: every other point, however, admits of a straight line being drawn, which meets and does not cut the curve in that point.

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When said to be right-angled, obliqueangled, obtuse-angled, acute-angled 2

(A) Of the first Properties, and of Triangles which are equal in all respects.

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(a) Any one of the angles of a triangle is less than two right angles cor. 5 (b) Triangles which have two sides and the included angle of the one equal to two sides, and the included angle of the other, each to each, are equal in all respects (c) Triangles which have two angles and the interjacent side of the one equal to two angles and the interjacent side of the other, each to each, are equal in all respects (d) If one side of a triangle be equal to another, the opposite angle is likewise equal to the angle opposite to the other; and conversely

(e) In an isosceles triangle,

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1. The angles at the base are equal to one another; and if the equal sides are produced, the angles upon the other side of the base are likewise equal cor. 6

2. The following lines, viz. the line which bisects the vertical angle, the line which is drawn from the vertex to the middle point of the base, and the line which is drawn from the vertex perpendicular to the base, coincide with one another cor. 7 3. The straight line which bisects the base at right angles passes through the vertex, and bisects the vertical angle cor. 7

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(f) Triangles which have the three sides of the one equal to the three sides of the other, each to each, are equal in all respects

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(n) Of triangles which have the two sides of the one equal to the two sides of the other, each to each, that which has the greater vertical angle has the greater base; and conversely (0) The three angles of a triangle are together equal to two right angles; and if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles 14

(p) In a right angled triangle

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cor. 14

1. One of the angles, viz. the right
angle, is equal to the sum of the
other two
2. The straight line which joins
one of the angles, viz. the right
angle, with the middle point of
the opposite side is equal to half
that side; and conversely, if a tri-
angle has either of these proper-
ties, it is a rightangled triangle
cor. 14

[(9) Any two triangles are equal to one another in all respects, when they have 1. Two sides and the included angle of the one equal to two sides and included angle of the other, each to each (b).

2. Two angles and the interjacent side (c).

3. The three sides (ƒ).

4. Two angles, and a side opposite
to one of them
cor. 14

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5. An angle of the one equal to an angle of the other, and the sides about two other angles, each to each, and the remaining angles of the same affection, or one of them a right angle (see II. 33.)] (r) Two right-angled triangles are equal to one another in all respects, when they have

1. The hypotenuse and a side of the one equal to the hypotenuse and a side of the other, each to each 10

2. The hypotenuse and an acute angle

3. The two sides (I. 4.)

10

4. A side and the adjoining acute angle (I. 5.)

5. A side, and the opposite acute angle (I. 19. and I. 4.)

(8) If in two right-angled triangles, the hypotenuse of the one is equal to the

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(B) Of the mutual Relations of the Sides. (a) In a right-angled triangle, the square of the hypotenuse is equal to the squares of the two sides (b) In every triangle, the square of the side which is opposite to a given angle is greater or less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and that part of it which is intercepted between the perpendicular, let fall upon it from the opposite angle, and the acute angle; greater if the given angle is greater than a right angle, and less if it is less

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(c) Any angle of a triangle is equal to or greater or less than a right angle, according as the square of the opposite side is equal to or greater or less than the squares of the containing sides

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cor. 21, 22 (d) In every triangle, if a perpendicular be drawn from the vertex to the base, the difference of the squares of the sides is equal to the difference of the squares of the segments of the base, i. e. the base is to the sum of the sides as the difference of the sides to the difference of the segments of the base or sum of the segments of the base produced (e) In a right-angled triangle, if a perpendicular be drawn from the right angle to the hypotenuse, the square of the perpendicular is equal to the rectangle under the segments of the hypotenuse, and the square of either side is equal to the rectangle under the hypotenuse and the segment adjoining to it; i. e. the perpendicular is a mean proportional between the segments of the hypotenuse, and either side is a mean proportional between the hypotenuse and the segment adjoining to it cor. 21 and 61 (f) In an isosceles triangle, if a straight line is drawn from the vertex to any point in the base, or in the base produced, the square of that straight line shall be less or greater than the square of the side by the rectangle under the segments of the base, or of the base produced

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(g) In every triangle, the squares of the two sides are together double of the squares of half the base, and of the straight line which is drawn from the vertex to the middle point of the base

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(h) In every triangle, if the vertical or

exterior vertical angle be bisected by a straight line which cuts the base, or the base produced, the base or base produced shall be divided in the ratio of the sides: also the square of the bisecting line shall be equal to the difference of the rectangles under the sides and the segments of the base, or base produced 70, 89 (i) In every triangle, if the base is equally produced both ways, so that the base produced is a third proportional to the base and sum of the sides, the sides of the triangle are to one another as the corresponding seg ments of the base produced sch. 64

(k) In every triangle, if the base is equally reduced both ways, so that the base reduced is a third proportional to the base and the difference of the sides, the sides of the triangle are to one another as the corresponding segments of the base reduced sch. 65 (1) In an isosceles triangle, if the equal angles are each of them double of the vertical angle, the sides and base are in extreme and mean ratio. See "Errata."

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Every triangle is a mean proportional between two rectangles, the sides of which are equal to the semiperimeter of the triangle and the excesses of the semiperimeter above the three sides.

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I

C

D

B

G

The demonstration of this is briefly as follows:Let a circle be described within the triangle ABC, and a circle upon the other side of B C, the one touching B C, A C, and A B, in the points D, E, F, and the other B C and AC, A B produced in the points G, H, K; then the centres L, M of these circles lie in the straight line which bisects the angle at A, and if B L, B M be joined, the angle LBM will be a right angle, because it is half the sum of the angles A B C, C B K; also, because A K and AH are together equal to A B, BG and AC, HCG, that is to the perimeter of the triangle ABC, each of them is equal to the semiperimeter. Again, because F B and B K (i. e. BD and BG) are together equal to E C and C H (i. e. to C D and CG), taking away G D from each, 2 BG is equal to 2C D, and consequently BG to CD; therefore F B is equal to C H, and BK to E C. A F, therefore, is the semiperimeter, B K is the excess above the side A B, B F the excess above the side A C, and A F the excess above the side B C.

K

M

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