and the scale of the original drawing, and the whole length, or any similar part of the required drawing are given.

PROB. IX. A drawing being given without a scale to proportionate another, having the dimension or extent of some part of the intended drawing. PL. 6. FIG. 4.

Draw two lines A B, B C, forming any angle A B C with each other, as before, from the angular point; on one of the lines B C set off the extent of the part of the required drawing, from B to C; from the same point B set the extent of the corresponding part of the other drawing, from B to A on the other line, and join A C. Make A B a scale of any number of divisions, as five, divide B C in the same proportion; sub-divide one of the extreme parts of A B into tenths, find the proportionate tenth of the corresponding part of B C; then will A B be a scale for the original drawing, and BC a corresponding scale for the required drawing.

Example, Figures 5, 6, 7.

Suppose A B C D A to be an original drawing, as a plate for a book, and to be of greater length or height than the page will admit of: then let the given height be E H, construct two proportional scales, fig. 7, as described in this problem, then all the dimensions and distances of the diagrams of fig. 6. will easily be proportioned to the corresponding dimensions and distances of the diagrams, fig. 5. A very accurate method, where any of the diagrams are very oblique, is to produce the sides to the boundary lines in the original drawing, then finding the corresponding points in the boundary lines of the required drawing, and by this means the angles of position may be had with the greatest correctness. In

circles, the position of their centres must be found by measuring from the corresponding boundaries, and then their radii from the respective scales. Parallel lines may be drawn by the parallel ruler.

20.

PROB. X. To draw a diagonal scale.

Suppose A B to be a scale agreed upon, consisting of 50 feet, the divisions separating each two adjacent 10 feet, being 0, 10, 20, Draw the parallel lines A C, . 0,. . 10, 10 .. 20, 20: 30, 30.. B D. Take any convenient opening of the compass, run ten parts from A to C, and from B to D, through the divisions, draw parallels; then C D being numbered as A B: divide A 0 into 10 equal parts, and also C 0; from the points 0, 1, 2, 3, 4, &c. in A B to the points 1, 2, 3, 4, &c. draw 0, 1; 1, 2: 2,3: 3, 4, &c. By this means you may obtain the hundreth part of the distance A 0, or C 0, according to the parallel you measure upon; thus, suppose you required 32 feet, and 4 tenths of a foot, you must place the foot of your compass on the fourth division from 30, on the line A B, in the vertical line 30, 30, and extend the other leg along the fourth parallel, till it fall upon the diagonal 2, 3, and this extent will be equal to 32.4 feet, and thus any extent whatever may be found.

Draftsmen seldom or never make use of a diagonal scale, as persons in the habit of drawing, will judge of any small part as nearly by the eye, as if measured by the best divided diagonal scale, at least without the assistance of a glass; and thus employ. ing a common scale will be a great saving of time. However, in the solution of a mathematical problem in mensuration, it may be applied with advantage where time would be of less consideration, in order to obtain the accuracy desired, or to confirm the truth of a calculation.

CARPENTRY.

§ 1. CARPENTRY in civil architecture, is the art of employing timber in the construction of buildings.

The first operation of dividing a piece of timber into scantlings, or boards, by means of the pit saw, belongs to sawing, and is previous to any thing done in carpentry.

§ 2. The tools employed by the carpenter are a ripping saw, a hand saw, an axe, an adze, a socket chisel, a firmer chisel, a ripping chisel, an augur, a gimlet, a hammer, a mallet, a pair of pincers, and sometimes planes, but as these are not necessarily used, they are described under the head of joinery, to which they are absolutely necessary.

§ 3 OF SAWS.

A saw is a thin plate of steel, indented on the edge, so as to form a series of wedges, with acute angles, and for the conveni. ency of handling, a perforated piece of wood is fixed to one end, by means of which the utmost power of the workman may be exerted in using it.

Saws have various names, according to their use. It is obvious, in order that the saw should clear its way in the wood, that the plate should decrease in thickness from the cutting edge towards