Chap. VIII.
GEOMETRY.
739
An Inaccessible Angle is formed by two lines which meet ina point, as A BC, the point of the pyramid B being supposedinaccessible, or which cannot be reached so as to measure it.
A solid Angle is the point where more than two planes orsuperficies of a solid touch each other. The point E is asolid angle between the three faces II, I, K: all points orangles of solid rectilinear bodies, of whatsoever figure, are solid,as more than two faces meet each other.
The cube is bounded by six squares, and constitutesone of the five regular Platonic bodies, which being placedbeside each other fill up the space about a point; there arehere eight solid angles formed by the junction of the sixplanes. The duplication of this solid, or the finding of theside of a cube, containing exactly twice as much as another,was for a long time a problem of difficulty, and which cannotbe solved by means of the straight line and circle, which werethe only lines the ancients made use of in constructing theirgeometrical solids.
The Opening , or Size of an Angle , is measured by the num-ber of degrees of the circumference of a circle containedbetween the two sides, the circle always having the summitof the angle for a centre. L 3M N is of GO degrees opening,because there arc so many contained between the lines LMand NM, or GO parts out of 360 -of the circumference, whichis described, taking M as a centre.
In geometry generally, the term right is applied to suchangles as have one line perpendicular to the other, aswhere the angle is one of 90 degrees. The Platonicschool of mathematicians was frequently employed to dis-cover rational numbers, which should designate the sides of
n 8 — 1
a right-angled triangle : Pythagoras gave the formula «—-—w 8 *t* 1
and n——, where n is odd: Plato gave 2 n, «*— 1, and « 8 + 1
where n is either odd or even. For practical purposes thenumbers 3, 4, and 5, effect this: suppose P Q, to be 4 feet,P O 3 feet, and the distance from Q to O to be 5 feet, then weknow that we have a right angle, for if to the square of 4 weadd the square of 3, and then extract the square root, weobtain 5.
The Angle O P Q is for the same reason one of 90 degrees,because the distance between O P and P Q is a quarter of thecircumference of the circle which encloses it.
A Right Angle is that made by a right line falling perpen-dicularly on another, and which contains in its opening aquarter of the circumference; A B C is a right angle, becausethe line AB falls perpendicularly on that of BC; a rightangle is square, and is used as such by all workmen.
The square G, formed with either wood or metal,enables us to set out very accurately a right angle, or anyother figure which has its sides perpendicular to the base ; fordrawing we have one limb fixed into a cross piece that givesit the form of a T: with such a square we can construct, bymeans of a drawing board nicely adjusted, lines both paralleland perpendicular.
The square used by mechanics is formed like an L, andshould be a true right angle; this can always be ascertainedby drawing a line along the edge of the blade, and then re-versing it; if the line and blade in the new position correspond,the square is pronounced to be true.
A rule or square, formed of wood or ivory, like a right-angled triangle, is used for drawing perpendicular lines; thisis laid with one side to the given line, and the perpendicularrequired is drawn by the edge of that at right angles to thefirst.
T squares are sometimes made of two equal pieces, kepttogether by a screw; the blade is fixed into one of these, flush
3 b 2
Fig. 641.
Fig. 643.
Fig 644.
Fig. 645.