4th. The regular Hexagon:-In this case each angle is equal to four-thirds of a right angle (S6 Cor. 1); .. three such angles are exactly equal to four right angles. Hence the regular hexagon is such a figure as is required.

This, in fact, follows from the first case, because each regular hexagon is made up of six equilateral triangles (160), as shewn by the dotted lines.

This combination of hexagons is remarkable as being the form adopted by bees in framing the honey-comb.

5th. Regular Polygons of a greater number of sides:Since each angle of a regular polygon evidently increases as the number of sides increases; and since three angles of a regular hexagon are equal to four right angles; .. three angles of every other regular polygon with a greater number of sides must exceed four right angles. Hence no other regular figures exist, for the purposes here required, except those already determined, viz. the equilateral triangle, the square, and the regular hexagon.

Cor. It follows from the first case that a plane area may be covered by lozenges, whose shorter diagonal is equal to a side; as is often seen in the glazing of windows.

211. PROP. CIX. To find what pairs of regular rectilineal figures, on the same base*, will exactly cover a plane surface.

Since each angle of an equilateral A = of a right angle, 1 right angle,

therefore,

a square a hexagon

an octagon

=

=

(1) 3 angles of A+ 2 angles of square = 4 right angles; (2) 4 angles of A+ 1 angle of hexagon 4 right angles; (3) 2 angles of A+ 2 angles of hexagon = 4 right angles; (4) 2 angles of octagon +1 angle of square = 4 right angles. These combinations of two figures will be represented as follows:

That is, the side of the triangle figures.

=

the side of each of the other

(1) The 1st thus:

(2) The 2nd thus:

(3) the 3rd thus:

(3) or thus:

(4) the 4th thus:

212. PROP. CX. To find what combination of three regular rectilineal figures, on the same base, will exactly cover a plane surface.

There appears to be only one such combination, viz. 1 angle of A +2 angles of square +1 angle of hexagon, the sum of which is exactly 4 right angles. This combination will appear thus:

This beautiful arrangement of three regular figures, although complex in appearance, may be constructed as a pavement with remarkable ease and certainty, by first marking down the dotted lines, as in the diagram; and may be extended readily to cover any extent of plane area whatever.

213. Of course by deviating from regular figures other arrangements may be made differing from those here given; for instance, the second case in (210) may be changed from squares to rectangles in an endless variety. So also in the 1st case of (211). And again, in the 4th case of (211), the octagons need not be equilateral provided they are equiangular, and equal to one another.

A very pleasing combination is made by adding to the 4th arrangement in (211) a narrow strip, as a border, to each octagon, composed of equal trapeziums, rectangular at one end, with the other two angles equal to one right angle and a half, and half a right angle, respectively; the shorter base of the trapezium being equal to the height of the trapezium + a side of the square.

PRACTICAL HINTS AND DIRECTIONS FOR YOUNG DRAUGHTSMEN.

THE Student will probably have gathered for himself, as he went along, many of the following notions; nevertheless it will not be without its use to recapitulate them here, in order that they may be the more strongly impressed upon his memory.

(1) See that you have good Tools to work withthat Rulers and Squares are correct-Compasses sharppointed and Parallel Ruler well tested.

(2) Recollect it is not an easy matter to draw a straight line perfectly straight. Flat-rulers are mostly feather-edged, to enable you to draw lines with ink, without blotting, by raising the edge above the paper. This is a source of error, where great accuracy is required. It is always better to use a pencil first, so that the edge, along which it is drawn, may be in contact with the paper. Use hard pencils, and keep their points fine.

(3) In using the Compasses hold them erect, move them with a gentle hand, and pierce the drawing paper as

little as possible. A large hole cannot be the centre of a circle, or of an arc of a circle.

(4) Never guess at a straight line which is required to be at right angles, or parallel to, a given line. And, mark, the shorter the line is, the less likely are you to guess right.

(5) Never use the end of the flat-ruler to draw a straight line at right angles to a given line from a given point in it. Use the ruler with cross-line at right angles to both edges.

(6) In using the flat-ruler, or parallel-ruler, be careful to hold it tight to the drawing paper with two fingers at least; for if one finger only be pressed upon the ruler, it will probably, and perhaps imperceptibly, revolve round that point to a small extent, as the pencil or pen is passed along its edge.

(7) Keep the joint of your compasses at a medium stiffness, neither so tight as to require much force in moving the legs, nor so slack as to prevent them from being handled without altering the angle between the legs.

(8) Use the compasses, where practicable, in preference to the flat-ruler or straight-edge, for the sake of accuracy. A point is in no way more correctly determined than by intersecting arcs, on which account constructions made with the compasses alone are generally most accurate. The only exception is, when the intersecting arcs make a very obtuse angle with each other, in which case the actual point of intersection is not easily detected.

(9) Draw all lines at first as long as they are likely to be wanted, with the pencil, whether straight lines or circular arcs. For a straight line is not so accurately 'produced' as it night in the first instance have been continued to the required extent. And it is best to continue circular arcs as far as they can be needed for two reasons: 1st, because, after once removing the compasses from the paper, you may miss the centre in the second attempt; and 2nd, because, if the compasses have been disturbed for any other purpose, or by accident, it may not be easy to hit the radius to a nicety.

(10) Never suppose, that you can produce' a very short straight line, or draw another parallel to it with even tolerable accuracy, by means of a flat-ruler only. Nothing can be more fallacious, unless you have previously determined some distant point through which the line is to pass.

(11) It is not an easy thing even to join two points accurately, when the points are very near to each other. In many cases there will be some other longer line in the construction, to which the shorter line is known to be parallel. If so, apply the parallel-ruler to the longer line, as a test line, and then force, as it were, the shorter line into parallelism with this.

(12) In constructions where the same straight line or length, is repeated often, it is well to have two, and sometimes, three, pairs of compasses in use, in order that, being once adjusted to their proper distances, they may accurately retain that adjustment throughout. This is especially necessary in the construction of regular polygons.

(13) Despise not the variety of methods given for doing the same thing; because that which is practicable in one case is not so in another under different circumstances. Recollect, for example, that theoretically any straight line may be 'produced'; but practically this is not true, because the line may already reach the edge of your paper, or your table, or room.

(1) WHAT is meant, when we say, 'join AB' and 'produce CD'?

(2) When a straight line is required to be drawn very accurately between two given points on a plane surface mention the several things which ought to be carefully attended to.

(3) If a straight line is to be drawn to join two given points and to be produced, is there a greater chance of error when the points are near together, or when they are at a considerable distance apart? and why?

PART II.

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