The diagonal scale is probably a centesimal scale, because by it an unit may be divided into 100 equal parts; and there fore any number, to the 100th part of an unit, may be expressed, which is an exactness generally sufficient in prac tical business. How this is done will he easy to understand, as follows: let a B, Figure Is, Plate Ill., be 1, or an unit, and divide it into 10 equal parts at 1, 2, 3, 4, &e. At a proper distance, B c, draw the line c n, equal and parallel to A B, and divide it also into 10 equal parts at a, b, c, d, &e : then join the points A a, I b, 2 c, 3 d, &e., and these will be the ten diagonal lines. Lastly, divide B C into ten equal parts also, and number them 1, 2, 3, 4, &.e., to 10 at c ; then, through each of these divisions, draw lines parallel to A through the length of the scale, and the construction is completed.
In this diagonal scale, A B is one inch ; then it' it be required to take off 1 inches, or 1.73, set one foot of the com passes in the third parallel under 1, at e, and extend the other fiat or point to the seventh diagonal in that parallel, at g ; and the distance e g is that required ; tor e f is one inch, and f g is 73 parts of 100.
Again, suppose it required to set off upon any line 2.37 inches ; then place one point of the compasses on the seventh parallel under 2, at 1, and extend the other to the third dia gonal in the same parallel at i ; and the distaiee it i is that reimired. Or, if A B be 10. the distance e g is 17.3, and it i is 23.7. Also, if A B be 100, then e g is 1 73,h i is 237 ; and so on.
This diagonal scale has the centesimal division at each end, and the unit in one is just the double of that of the other : thus, it' A 13 be one inch at one end, it is half an inch at the other ; or it' it he half an inch in the larger, it is one quarter in the lesser divisions, as is the case upon most of the common plain scales.
This unit, A B. may also lie one foot, one yard, one rod, one mile, &c. So that every unit in every kind of measure is hereby estimated in hundredth parts of the whole, which shows the diagonal scale to he a most useful invention.
On the other side of the plain scale are the seven decimal lines, which are usually called plotting scales, because, their divisions of an unit into ten parts being different in the pro portion of 4 to I, the surveyor may vary the scale of his plot or plan of an estate. &c., in that ratio, in seven different ; and the superficies or sizes of the greatest and least plans, will be as 16 to 1. Or, that drawn by scale No, 10, will lie sixteen times larger than the plan laid down from scale No. 40.
The same variety is also to be had in the construction of all other geometrical figures, whether superficies or solids ; and with respect to the latter, the greatest will be the least as 64 to 1 ; that is, the architect can vary the size of his strue ture in the ratio of 64 to 1, in seven different elevations.
The last line on the common plain scale is that of chords; and is much more used than the protractor for laying off or measuring any proposed angle. Thus, let it be required to draw the line a C. Figure 19, to make an angle of 35° with the line A c. To do this, set one point of the compasses in the beginning of the line of chords, and extend the other to 60 ; with that extent, as a radius, place one foot in C, and with the other describe the are A B ; then take from the chords 35° in the compasses, and set them on the arc from A a ; then through C and a draw c a, and it is done.
Again.-If it be required to measure any angle, as A B Figure 20, produce n A, it' necessary ; take 60° from the chords in the compasses, and with one foot in a describe the arc A C, emitting the leg a A at A, and a C at C ; then take the are A C 111 the compasses, and applying it upon the beginning of the line of chords it will reach to 30', the quantity of the angle required. But the line of' chords is more useful on the sector, to which we now proceed.
N.B.-The construction of this line is shown in the next article.
The sector is a most useful instrument, as forming a uni versal plane scale.
The lines commonly laid down upon the sector, are-aline of equal parts, marked L at the end ; a line of chords to 60°, marked c ; a line of sines, to 90°, marked s ; a line of tan gents, to 45°, marked T ; another line of tangents from 45 to '75 or upwards, marked t a ; a line of secants, marked s e; a line of polygons marked r o a.
Besides these, when the sector is quite opened, there arc placed on one side-a Gunter's line of artificial numbers, n; a line of artificial sines, s ; and a line of artificial tangents, t. Likewise a line of 12 inches, and another of the foot divided into 1000 equal parts, placed by it, for the purposes already Before a proper idea can be formed of these sectoral lines, and their uses, their construction must be shown from the circle. Therufbre. let A C B, Figure 21, be a quarter of a circle, divided into 90", described with the radius c on the centre c ; let A E and c F he perpendicular to A C, At A and C; then if the radius A C be divided into 10, 100, 1000, Sze., equal parts. it will lie the line so called upon the sector.