3. To find the value of 3 inches. 4. To find the value of .15 foot.

Ans. 25 foot.

Ans. 1 inch.

The Diagonal Scale is likewise a centesimal scale, for by it an unit is divided into 100 equal parts; and any number of those parts may be taken in the compasses, and laid down on paper with sufficient exactness for most practical purposes.

To explain the construction and use of the Diagonal Scale, let ABCD be a section of the scale, which is equally divided (suppose into inches) from B towards A in the points E, 1, 2, 3, &c. Let BC BE: and let each of these be divided into 10 equal parts in the points marked by the small figures, 1, 2, 3, 4, &c. I, II, III, IV, &c. also divide CF in the same manner in the points a, b, c, d, &c. and let the lines passing through B, E, 1, 2, 3, be perpendicular to AB, and the lines kl, nII, mIII, oIV, &c. parallel to it, join 9 C, 8 a, 7 b, 6 c, 5 ́d, &c.

Since 9 B=BI=aC, and 9 C by its inclination to BC meets it in C, if the distance of 9C and BC at B, that is 9B, be called 1, then will their distance on the next parallel marked I be

8

9

and at the next parallel marked II, it will be

10'

; at the

next marked III, it will be

7 10

; at the next marked IV, it will

6

;

10

1

be and so on, decreasing successively by down to the 10'

point C, where the lines meet, and consequently the distance is nothing.

If 8B be called 2, then will the distance from 8 a to BC on the parallel marked I, be 1; on the parallel marked II, 1; on the parallel marked III, 17; on the parallel marked IV, 1; and the like for other divisions.

EXAMPLES.-1. Let it be required to find 3.4 on the scale.

Here it will be convenient to begin at E; wherefore if the distance of the lines EF and 3 f be taken in the compasses on the parallel marked IV, it will be 3.4, the number required.

2. To find 7.8 on the scale.

Extend the compasses from EF to 7b on the parallel marked VIII, and it will be the distance.

3. To find 3.45 by the scale.

In this case we must take each of the primary divisions marked with the large figures, 1, 2, 3, &c. for unity, and then the smaller divisions, E 1, &c. will each represent one tenth, and the parallel differences each one hundredth; wherefore we must extend the compasses from 3 D to 4e on the parallel marked V, and it will be the distance required.

The Diagonal Scale has the decimal and centesimal division at each end, the unit of one being double that of the other, for the convenience of drawing figures of different sizes.

The other side of the Plain Scale contains seven lines decimally divided and subdivided; these are called Plotting Scales, and serve to construct the same figure of seven different sizes: by the help of these we can accommodate the figure to the dimensions of the page or sheet on which it is required to be drawn. The number at the beginning of each of these lines shews how many of its subdivisions make an inch.

The line of chords marked C on this side of the Plain Scale, is used for the same purposes as the Protractor, viz. to measure and lay down angles, &c. The method of using both will be explained hereafter.

e The method of diagonals was invented by Richard Chanseler, an Englishman, and first published by Thomas Digges, Esq. in his Ala, seu Scale Mathematica, London, 1573.

The SECTOR is an instrument consisting of two flat rulers or legs, moveable on a joint or axis, the middle point of which is the centre; it contains all the lines usually set on the Plain Scale, and several others, which the peculiar construction of this useful instrument renders universal.

The lines on the Sector are distinguished into two kinds, single and double.

The single lines on the best Sectors are as follow;

1. A line of Inches decimally divided.

2. A line of a Foot centesimally divided on the edge.

3. Gunter's line of the Logarithms of Numbers, marked n 4. Logarithmic Sines.

Tangents to 45 degrees
Secants

Tangents above 45 degrees
Polygons.

f The first printed account of the Sector appeared at Antwerp in 1584, by Gasper Mordente, who says that his brother Fabricius invented the Sector in the year 1554. Some ascribe the invention to Guido Ubaldo, A. D. 1568: others again to Justus Byrgius, a French mathematical instrument maker, who also flourished in the 16th century. Daniel Speckle next treated of the Sector, viz. at Strasburg in 1589, and Dr. Thomas Hood wrote on the same subject at London in 1598, as did Samuel Foster, in a posthumous work published at London by Leybourne, in 1661. Many others have since explained the nature and uses of this instrument; but the most complete account of any will be found in Mr. Robertson's Treatise of Mathematical Instruments.

The scales of Lines, Chords, Sines, Tangents, Rhumbs, Latitudes, Longitudes, Hours, and Incl. Merid. being set on one leg only, may be used with the instrument either shut or open. The scales of Inches, Decimals, Log. Numbers, Log. Sines, Log. Versed Sines, and Log. Tangents, are on both legs, and must be used with the instrument open to its utmost extent.

The double lines proceed from the centre or joint of the Sector obliquely, and each is laid twice on the same face of the instrument, viz. once on each leg. To perform operations peculiar to the Sector, or, as it is called, "to resolve problems sectorwise," its legs must be set in an angular position, and then distances are taken with the compasses, not only "laterally,” (or in the direction of its length,) but "transversely," or " parallelwise," viz. from one leg to the other.

The PARALLEL RULER consists of two straight flat rules, connected by two equal brass bars, which turn freely on four pins or axes, fixed two on each rule at equal distances, so that the rules being opened, or separated to any distance within the limits of the bars, they will always be parallel, and consequently the lines drawn by them will be parallel.

The BLACK LEAD PENCIL should be made of the best black lead, and its point scraped very fine and smooth; it is used for drawing lines by the edge of a scale or ruler where ink lines are not wanted. Plans and figures which require exactness, should be first drawn with the pencil, and then if they are not right, it will be easy to take out the faulty part with a piece of India rubber, and make the necessary correction; after which the pencil lines may be drawn over with ink. The pencil is not less convenient as a substitute for the pen in writing, calculating, &c. A piece of good clean India rubber, of a moderate size and thickness, must always accompany a case of Mathematical Instruments.

8 The Parallel Ruler usually put into a case of Instruments is only six inches long, and too small for most purposes; the better sorts are from six inches to two feet in length, and sold separate.

The Double Parallel Ruler consists of three rules, so connected that the two exterior rules move not only parallel, but likewise opposite to each other; for some account of its construction and use see Martin's Principles of Perspective, p. 28.

The foregoing short description was deemed necessary, but the uses of the Instruments must be deferred, until the learner has acquired sufficient skill in Geometry to understand them.

OF GEOMETRY, CONSIDERED AS THE SCIENCE OF DEMONSTRATION.

As the reader is supposed to be unacquainted with logic, it will be proper in this place to introduce a few particulars taken from that art, which may serve as an introduction.

1. The mind becomes conscious of the existence of external objects by the impressions it receives from them. There are five inlets or channels, called the organs of sense, by which the mind receives all its original information; namely, the eye, the ear, the nose, the palate, and the touch: hence seeing, hearing, smelling, tasting, and feeling, are called the five senses. This great source of knowledge, comprehending ali the notices conveyed to the mind by impulses made by external objects on the organs of sense, is called SENSATION.

2. PERCEPTION is that whereby the mind becomes conscious of an impression; thus, when I feel cold, I hear thunder, I see light, &c. and am conscious of these effects on my mind, this consciousness is called perception.

3. AN IDEA results from perception; it is the representation or impression of the thing perceived on the mind, and which it has the power of renewing at pleasure.

4. The power which the mind possesses of retaining its ideas, and renewing the perception of them, is called MEMORY; and the act of calling them up, examining, and reviewing them, is called REFLECTION.

5. In addition to the numerous class of ideas derived by sensation wholly from without, the mind acquires others by reflection; thus by turning our thoughts inward, and observing what passes in our own minds, we gain the ideas of hope, fear, love, thought, reason, will, &c. The ideas derived by means of sensation are called SENSIBLE IDEAS, and those obtained by reflection, INTELLECTUAL IDEAS.

6. From these two sources alone (viz. sensation and reflection) the mind is furnished with ample store of materials for its future operations; sensation supplies it with the original