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perficies, just as they would appear to the sight, if seen through a transparent plane, a pane of glass, or window.

In the representation of solid bodies, buildings, &c., there are three divisions :

1. Ichnography, which shows the plan or groundwork of the building.

2. Orthography, which exhibits the front or parts in direct view.

3. Scenography, which is the perspective view of the whole building, fronts, sides, and height.-See Drawing, &c.

Obs. 1.-Sciagraphy, or dialling, is the art of making dials on all kinds of planes ; as horizontal, erect, or declining, or erect and reclining. The hour-lines, the height of the stile or gnomon above the plane, the distance of the substile from the meridian, and the difference of longitude, are all calculated by spherical trigonometry.

2. In a work like the present, correct general views are all that can reasonably be expected, and the details of the common sciences of reading and writing, grammar, arithmetic, and book-keeping, are supposed to be acquired in the routine of school-business, from such common books as Blair's English Grammar, Joyce's Arithmetic, Morrison's Book-keeping, and Nicholson's Popular Elements of Mathematics.

XXV. Algebra ; or Abstract Arithmetic. 574. If, in calculations, we were to substitute letters for known numbers, and operate with them by the signs +, -, X, and ;-, till, by reasoning, we have acquired such a disposition of the said letters as expresses the result, and nothing but the result, we should simplify and shorten the calculation.

Such, then, is the science called Algebra. We adopt any letters of the alphabet at pleasure in place of any

given numbers, and operate with them by the intervention of signs till we have the result.

Obs. For the sake of precision, it is usual to take the first letters for any known numbers, as a, b, c, d, &c. and the unknown numbers from the last letters as X. y. s.

575. The algebraic signs are as follows: + More or add as atb is a more b, or a added to b. -Less; as amb, is a a less by b.

xMultiplied : as axb, is a multiplied by b; or ab without the sign, is the same.

-Divided; as a-;b, is a divided by b; or a is the same and more usual.

b =Equal to; as a=b is a equal to b.

: is to; as thus, a: b::c:d; that is, as a is to b, :: So is: so is c to do

Q Involved; as squared, cubed, &c. w Evolved; or the root extracted.

The root; as v ab, is the square root of a b. The power, as aor b3 is the square of a or the cube of b; that is a X a, or b xbxb.

576. When an arithmetician wishes to perform his problems algebraically, he writes down the data of his question, as severally equal to a, or b, or c, as he pleases; and the unknown or sought members, as x or y, or z; and then adds, subtracts, multiplies, and divides these by the signs, as his reason directs till he arrives at a simple result.

Obs. Suppose the joint ages of Eliza and Emily are 23, and that Eliza is three years older than Emily, and their respective ages are required: put a then= 23 and b=3 and call Eliza's unknown age y, and Emily's z, and then algebraically it will stand,

a=y+

b=y-2. Then, if these are added together, as tz added to m2 destroys itself, it will then stand a+b=2 Y; which, divided by 2 gives

a+b

Eliza's age;

2

23+3 and by restoring a and b, it will standy, or 26

2 13, Eliza's age, and Emily's being 13-3

2 will of course be 10.

2. A child, to whom the four first rules of arithmetic and the characters are known, may be made to understand this ; and I advise no female or young person to pass it as a difficulty. A course of Algebra may be undertaken, after a youth is master of vulgar and decimal fractions.

577. Fluxions are the different velocities, or moving forces, whereby any quantities, in a flowing state, increase or decrease, according to the ratios of the velocities.

Quantities and their fluxions are (as in algebra,) represented by letters ; known quantities, by the first letters, a, b, c, d, &c. ; and the fluents, by the last, as v, x, y, z; and their fluxions, by the same with a point over them, as v, x, y, z.

Obs.--As the ratios of velocities, in many cases, are perpetually altering, as in the motion of falling bodies, these fluxions vary every moment, and produce fluxions of fluxions, or second fluxions, thus marked, v, X, Y, Z, and the fluxions of these

are third fluxions, as v, X, Y, Z.

XXVI. Optics. 578. This science is founded on the properties of Light, which derives its chief source from the Sun, and is also generated or decomposed by bodies in a state of combustion

MALLET.

We ascertain the utility of light, by the introduction of a candle or ray of sun-shine into a dark room. This, in an instant, renders every thing visible by the emission of innumerable rays, or particles proceeding from the candle or ray, to the objects, and from them to the eye, producing therein a figure of the objects ; and a corresponding sensation in the brain.

Fairest of beings! first created Light!
Prime cause of beauty! for, from thee alone
The sparkling gem,--the vegetable race,
The nobler worlds that live and breathe, their charms,
The lovely hues peculiar to each tribe, --
From thy unfading source of splendour draw!
In thy pure rays, with transport I survey,
This firmament, and those her rolling worlds ;--
Their magnitudes and motions,

Obs.-Two hypotheses have been invented to account for the principal operations of light. In the first, it is supposed, that the universe contains a highly rare elastic substance, which when put into a state of undulation, produces those effects on our organs of sight, which constitute the sensations of vision, and the other phænomena occasioned by solar and terrestrial rays. In the second, it is conceived, that particles are emitted, or sent off, from luminous or heatmaking bodies with great velocity, and that they produce their effects by communicating their motions to substances, or by entering into them, and changing their composition. The first of these suppositions was adopted by Hook, Huygens, and Euler; the second, by Newton, and the philosophers of the Newtonian School. Most of the phenomena may be accounted for, by either hypothesis ; but the Newtonian doctrine applies more happily to some of the facts discovered, respecting the modifications of light by double refraction and reflection.

579. By observing the regular eclipses of Jupiter's moons in different parts of the earth's orbit, it is found, that rays, or vibrations of light, travel twelve millious of miles in a minute; yet, they do not affect the eye in passing into it; and they could never be found to produce the slightest impression on the most delicate balance.

It requires 12 rays or pulsations of light to fall on the eye in every second, to produce a constant perception of the object, whence the rays or pulsations proceed.

Hence rays, or pulsations of light, in passing from distant bodies to the eye, may be 16,000 miles behind each other, and yet produce constant vision.

Obs.The eclipses of Jupiter's moons are calculated for a mean distance of the earth ; but they happen sooner or la. ter, as the earth is nearer or more distant from Jupiter. The number of rays necessary to produce vision, is ascertained, by turning a piece of burning wood in a circle till the circle is wholly illuminated. The twinkling of the stars, doubtless, arises from paucity of rays.

580. Pulsations of light pass freely through air, water, glass, the coats and humours of the eye, and other transparent mediums. At the back of the eye, is spread a net of nerves, called the optic nerve ; to receive their impression, and communicate their effect to the brain.

The rays pass through a small hole called the pupil, and form, on the optic nerve, a beautiful and perfect picture of the objects before the eye. A camera obscura acts on the principle of the eye : and a common spectacle-glass will shew the same effect, held at a proper distance, from a wall.

581. But though effects of light pass in straight lines through any medium when in it; they are turned out of their course, as they pass obliquely out of one transparent body into another; and this effect is called refraction.

If a stone be thrown obliquely into water, it will be evident that when it strikes the water, it will fall to the bottom in a direction more perpendicular than before it came into contact with the water. Such, too, is the

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