of the paper, produce A B, C D, to meet it at H and T, and by the preceding problems divide HT at M, N, P, Q, similarly to A C; then the lines E M, F N, G P, and R Q, will give the bottoms and tops of the windows.

DIVISIONS OF LINES INTO EQUAL PARTS.

110. The mode of bisecting a line has already been given, and by continued bisection a line can be divided into 4, 8, 16, &c., equal parts.

111.-PROBLEM 36.-To trisect a given straight line A B. METHOD I.-On A B Fig. 82, describe the equilateral triangle

ABC; bisect the angles at A and B by lines intersecting in the point D, and through D draw parallels to CA and C B cutting A B in the points E and F; then will A B be divided at E and F into three equal parts.

Reason. The triangle DEF is equiangular (Theor. 9) to the triang'e A B C, and therefore equilateral (Theor.4); the angle

ADE is equal to the angle DA C (Th or. 8), and, consequently,

to the angle D A E; therefore A E is equal to ED, that is to EF; and in like manner BF is also equal to E F. METHOD II.-Draw AD, Fig. 83, making any angle with AB; and set off on it equal distances, A C and CD; join BD, and produce it, making D E equal to DB; through E and C draw a straight line, cutting A B in F; and through D draw a parallel

Fig. 83.

to E F, cutting A B in G; then A B will be trisected in the points F and G.

Reason.-CF being parallel to the side DG of the triangle ADG, AF is to FG as AC to CD (Theor. 15); but AC is equal to CD; therefore also AF is equal to F G. Again, GD being parallel to FE, a side of the triangle BFE, FG is to GB as ED to DB; but ED is equal to DB; therefore also F G is equal to GB; therefore AF, FG, and GB, are equal to each other, and the line AB is trisected in F and G.

METHOD III.-From A and B, Fig. 84, draw A C and BD parallel to one another; on these lines set off A E, E C, BF, FD, all equal to each other:

join DE, and CF, intersecting A B in the points G and H; then AB shall be divided into three equal parts in the points G and H.

Reason.-D F and EC being equal and parallel straight lines, D E and C F are also (Theor. 10) equal and parallel, and therefore (Theor. 15) A G is to G H as AE to EC; and as AE is equal to E C, A G

Fig. 84.

is also equal to G H. In like manner B H is equal to H G; and, therefore A G, G H, and H B, are equal to one another, and A B is trisected in the points G and H.

EXERCISES.

(1.) Draw a line 2 inches long, and trisect by Method I. (2.) Draw a line 24 inchs long, and trisect it by Method II. (3.) Draw a line 1.7 inches long, and trisect it by Method III.

D

112. PROBLEM 37.-To divide a straight line into any number of equal parts.

METHOD I.-Let A B, Fig. 85, be the given line; draw a straight line A C, making any angle with AB; take any distance, and set it off along A C, from A to C, as many times as the number of parts into which AB is to be divided; through the divisions thus marked on AC draw parallels to BC, intersecting A B; then A B will be divided into the number of equal parts required. Reason.-A B is divided similarly to A C (Theor. 17).

Fig. 35.

METHOD II.-Draw

A C, Fig. 86, making any angle with A B as before; set off on A C, from A to C, any distance as many times less one as the

Reason. Let C G be the last of the parallels to DFE; then (Theor. 17) A G is divided into the same number of equal parts as AC. Again, the triangle ADF being equiangular to the triangle BG C, and the side AD equal to the side B C, B G is also equal to A F, one of the equal parts into which A G has been divided.

EXERCISES.

(1.) Take a line 3.3 inches long, and divide it into seven equal parts by Method I.

(2.) Take a line 2.74 inches long, and divide it into five equal parts by Method II.

CONSTRUCTION OF SCALES.

113. To represent within a limited compass plans or drawings of objects of a considerable extent, and to be able to estimate the real dimensions of the objects represented from the drawings representing them, it is necessary to take for the representation of each unit of linear measurement, a length bearing the same proportion to the unit represented, as the linear dimensions of the surfaces, on which the drawings are to be constructed, bear to those of the objects. Thus if it be desired to construct within a width of 10 inches, a map of a tract of country whose greatest width is 45 miles, each mile of actual distance must be represented on the paper by a length of of an inch, and consequently each furlong by 3 of an in h. If then a line two inches long be divided into nine equal parts,

and one of these parts be subdivided into eight equal parts, and the primary divisions be taken to represent miles, and the subdivisions, furlongs, a scale will be formed, by means of which the representative distances between the points to be laid down on the map can be set off, and the map constructed; and again, by means of this scale, the actual distance between any two points represented in the map can be determined.

114. Such a scale as we have just described, in which two inches represent nine miles, is called a scale of 1 inch to 4 miles, or of of an inch to a mile.

Also since 4 miles contain 285,120 inches, and this distance is represented on the scale by 1 inch, it is called a scale of 288120, and 283120 is called the representative fraction.

115. Since the paper on which the drawings are constructed expands in moist weather, and contracts in dry, the scale upon which they are constructed should be laid down on the paper itself, so as to expand and contract with it. By this means whatever changes may take place in the hygrometric state of the atmosphere during the operation of constructing the drawinge, all their points will be laid down in their proper relative positions, at their proper representative distances from each other, and the scale will at all times determine with accuracy the real distances between the points so laid down.

116. The ordinary or simply divided scales are most usually subdivided decimally, that is, a certain length being divided

into a number of equal parts, or primary divisions, the primary division at one extremity of the scale is subdivided into ten equal parts; so that each subdivision represents one-tenth of the distance represented by a primary division, or a primary division represents ten times the length represented by a subdivision.

117. On such decimally subdivided scales, the dimensions are taken off to two places of figures, which may be tens and units, or hundred and tens, or units and tenths, or any other two orders of figures, of which a figure in the one is ten times the value of the same figure in the other.

118. Simply divided scales are, however, frequently so subdivided as to designate two orders of units of measurement, miles and furlongs, feet and inches, &c. In this case the primary divisions represent the higher order of units, and at one extremity of the scale a primary division is subdivided into the same number of equal parts as the number of units of the lower order contained in one unit of the higher.

119. In constructing scales it must be borne in mind that by taking a short measurement and repeating it, the error in the measurement is multiplied in the whole length set off by as many times as the measurement is repeated; while by taking a long measurement and dividing it, the error in the measurement is divided among the equal parts thus formed. Such a length, therefore, as will suffice for the whole or the greater part of the scale should be taken, and divided first for the primary divisions; and the division at one extremity, usually at that on the left hand, should be subdivided as already explained.

120. In figuring the scales, the zero is not at the extremity of the scale, but at the end of the first or subdivided primary division, and the figures run from this point in one direction for the primary divisions, and in the other for the subdivisions.

EXAMPLES OF SIMPLY DIVIDED SCALES.

121. The following examples will illustrate the mode of proceeding.

(1.) To construct a scale to measure single yards, in which 109 yards shall be represented by 6 inches.

Since the subdivisions are to represent single yards, the primary divisions must represent ten yards each, and the total