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after the manner of quicksilver, it would reqnire a tube 36 feet long or more; which could hardly find room within doors: but then it would go 14 times more exact than quicksilver; because, for every inch tie quirk. silver rises, the water would rise 14, from whence every minute change in the atmosphere would be discernible.
And the water-barometer above described will show the variation of the air's gravity as minutely as the other, if the bottle be large, to hold a great quantity of air. And, in any case, by reducing the botule (so far as the air is contained,) to a cylinder; and put D = diameter of the bottle, d= diameter of the pipe, p= height of air, x= rising iu the pipe, all
408dd in inches. Then the height of a hill in fect will be nearly 1 +
408dd x 713. And if y = height of the hill or any ascent, Q=
very near, at a mean density of the air.
Fig. 18, is a Thermometer, or au instrument to measure the degrees of heat and cold. AB is a bollow tube near two feet long, with a ball at the bottom: it is filled with spirits-of-wine, mixed with cochineal, halfway up the neck; which done, it is beated very much, till the liquor fill the tubc; and then it is sealed hermetically at the end A: then the spirit contracts within the tube as it cools. It is enclosed in a frame, wbich is graduated into degrees, for heat and cold: for hot weather dilates the spirit, and makes it run further up the tube ; and cold weather, on the contrary, contracts it, and makes it sink lower in the tube. And the particular divisions show the several degrees of heat and cold; against the principal of which the words heat, cold, temperate, &c. are written.
*** Those who are desirous of pursuing the delightful and useful strely of MECHANICS, may consult Dr. Gregory's Treatise, where the inquisitive Reader will find this branch of Mathematics, with all its recent improremenis, treated, not only in the most scientific manner, but so as to erplain the prisciples, and illustrate the best practical modes for putting them in execution, by descripëive examples of machinery,
I. On the Nature of Light, and the Laws of Reflection and
Refraction. By Optics, we understand that branch of natural philosophy which treats of the nature, properties, and accidents, of light, and the theory of vision.
Modern philosophers have invented two hypotheses to explain the manner in which vision is produced by luminous objects.
Des Cartes, Huygens, aud Euler, suppose that there is a subtile elastic medium, which penetrates all bodies and fills all space; and that vibrations, excited in this fluid by the luminous body, are propa. gated thence to the eye, and produce the sensation of vision, in the same manner that the vibrations of the air, striking against the ear, produce the sensation of sound.
The other hypothesis, adopted by Sir I. Newtou and his followers, states, that light consists of very small particles of matter, which are constantly thrown off from luminous bodies, and which produce the sensation of vision by actual impact upon the proper organ.
Chemistry, and the actual phenomena of Combustion, qualify the first hypothesis, and impeach the second; but, though we may, in many respects, be ignorant of the actual operation of light, we inay, in reasoning upon it, adopt the notion that it consists of distinct and independent parts.
Definitions.-1. The least portion of light, which may be stopped alone, or propagated alone, or does or suffers any thing which The rest of the light does not or suffers not, is called a ray of light.
Rays of light may be represented by lines, drawn in the directions in which the particles move, or are affected.
2. Whatever affords a passage to the flow of light is called a medium, as glass, water, air, &c.; and, in this sense, a vacuum is called a medium.
3. The density of light is measured by the number of parts, or atoms, or particles, uniformly diffused over a given surface.
Cor.-If the surface be not given, the density varies as the number of pa ticles directly, and inversely as the area over which they are diffused.
There is something extremely subtile in the nature of light; and its properties can with difficulty be explained, either on the supposition of its materiality, or on that of its being only an accident of an elastic medium. The facility and regularity with which it is transmitted through bodies of considerable density, cannot be accounted for on either lıypothesis. If it colisist of particles of mat. ter, which is the Newtonian supposition, their minuteness greatly exceeds the limits of our faculties, even the power of human imagination. Noiwithstanding the astonishing velocity of these par
ticles, their momentum is not so great as to discoinpuse the delicate texture of the eye; and, when they are collected in the focus of a powerful burning-glass, it seems doubtful whether they are capable of communicating motion to the thinnest lamina of metal that can be exposed to their impact.
Prop. 1.- A ray of light, whilst it continues in the same uri. form medium, proceeds in a straight line
For, objects cannot be seen through bent tubes; and the shadows of bodies are terminated by straight lines. Also, the conclusions, drawn from calculations made on this supposition, are found by experience to be true.
PROP. 2.-The density of light varies inrersely as the square of the distance from a luminous point; supposing no particles to be stopped in their progress.
For, if the point from which the light proceeds be considered as the common centre of two spherical surfaces, the saine particles, which are uniformly diffused over the first, will afterwards be diffused, in the same manner, over the latter; and, since the density of light varies, in general, as the number of particles directly, and inversely as the space over which they are uniformly diffused, in this case it varies inversely as the space over which they are diffused, because the number of particles is the same; therefore, the density at the first surface : the density at the latter :: the area of the latter surface : the area of the former; that is, :: the square of the distance in the latter case : the square of the distance in the former.
Definitions.-1. When a ray of light, incident upon any surface, is turned back into the medium in which it was moving, it is said to be reflected.
2. When a ray of light passes out of one medium into another, and has its direction changed at the common surface of the two mediums, it is said to be refracted.
3. The angle contained between the incident ray and the perpendicular to the reflecting, or refracting, surface at the point of incidence, is called the angle of incidence.
4. The angle contained between the reflected ray and the per. pendicular to the reflecting surface at the poiut of iucidence, is called the angle of reflection.
5. The angle contained between the refracted ray and the perpendicular to the refracting surface, at the point of incidence, is called the angle of refraction.
6. The angle contained between the incident ray produced, and the reflected or refracted ray, is called the angle of deviation.
If RS (fig. 1,) be the reflecting surface, AC a ray incident upon it, CB the reflected ray, and PCQ be drawn, through C, perpendicular to RS, and AC be produced to E; then ACP is the angle of incidence, PCB the angle of reflection, and BCE the angle of deviation.
If RS be a refracting surface, and CĎ the refracted ray, then QCD is the angle of refraction, and ECD the angle of deviatiou.
PROP. 3.-The angles of incidence and reflection are in the same plane, and they are equal to each other.
Let a ray of light AC, (fig. 2,) admitted through a small hole into a dark chamber, be incident upon the reflecting surface RS at C; and let CB be the reflected ray; draw CP perpendicular to the reflector. Then, if the plane surface of a board TS he made to coincide with CA and CP, the reflected ray CB is found also to coincide with the plane TS; or the angles of incidence and reflection are in the same plane.
Again, if from.C as a centre, with any radius CA, the circle RPS be described, the arc AP is found to be equal to the arc PB; therefore, tho angle of incidence, ACP, is equal to the angle of reflection, BCP.
The angles of incidence and reflection are also found to be equal when rays are reflected at a curve surface,
Cor. 1.–The angles ACR, BCS, which are the complements of the angles of incidence and reflection, are also equal.
Cor. 2.-If BC be the incident ray, CA will be the reflected ray. For, the angle PCA is equal to the angle PCB, and in the same plane; therefore CA is the reflected ray.
Cor. 3.- If the ray PC he incident perpendicularly upon the reflecting surface, it will be reflected in the perpendicular CP.
Cor. 4.- If AC be produced to E, the angle BCE, which measures the deviation of the ray AC from its original course, is 180° ._ACB; or 180° – 2 % of incidence.
Cor. 5.-A ray of light will be reflected at a curve surface, in the same manner as at a plane which touches the curve at the point of incidence.
For, the angle of incidence, and conseqnently the angle of reflection, is the same, whether we suppose the reflection to take place at the curre, or the plane.
PROP. 4.—The angles of incidence and refraction are in the same plone; and, whilst the mediums are the same, the sine of the angle of incidence is to the sine of the angle of refraction, in a given ratio.
Upon the surface of a board TV, (fig. 3,) with the centre C and any radius CA, describe a circle PRQ, draw the diameters RS, PQ, at right angles to each other, and immerse the board into a vessel of water, in such a manner that PQ may be perpendicular to, and RS coincide with, the surface of the water. Then, if a ray of light, admitted through a small hole into a dark chamber, be incident upon the surface RS in the direction AC, coincident with the plane of the board CB, the direction of the refracted ray is found to coincide with that piane; that is, the angles of incidence and refraction are in the same plane.
iso, if Au and BF be drawn at right angles to PQ, they are the sines
incidence and refraction to the radius CA; and it is found that AD bas 80 BP the same ratio, whatever be the inclination of the incident ray to tire refracting surface. That is, if aC be any other incident ray, Co the refracted ray, ad and bf the sines of incidence and refraction, then AD: BP :: ad : bf.
The ratió of the sines of incidence and refraction is the same, when the refracting surface is curved.
Cor. 1.-Hence, if the angles of incidence of two'rays be equal, the angles of refraction are also equal.
Cor. 2.-As the angle of incidence increases, the angle of refraction increases.
Cor. 3.-A ray of light is refracted at a curve surface, in the same nianner as at a plane which touches the curve at the point of incidence.
Prop.5.- If a ray AC be refracted at the surface RS in the direction CB, then.a ray BC, coming the contrary way, will be refracted in the direction CA.
The construction being made as before, let a small object be placed upon the board at B, (fig. 4;) and, when it is immersed perpendicularly in water, till RS coincides with the surface, the object B will be seen from A, in the direction AC; and, since the motion of ligbt, in the same nurdium, is rectilinear, the ray, by which the olject is seen, is incident at C, and refracted in the direction CA.
Cor. 1.--The angle of deviation of the ray AC, is equal to the angle of deviation of the ray BC, wbich is incident in the contrary direction.
Cor. 2.- When a ray of light passes out of air into water, the sine of incidence : the sine of refraction :: 4 :3; consequently, when a ray passes out of water into air, the sine of incidence : the sine of refraction :: 3:4.
In the same manner, out of air into glass, the sine of incidence : the sine of refraction :: 3 : 2; therefore, out of glass into air, the sine of inci dence : the sine of refraction :: 2:3.
ScHolium. The preceding propositions, which are usually called the Laws of Reflection and Refraction, are the principles upon which the theory of vision is founded. They were discovererl, and their truth bas been established by repeated experiments, made expressly for this purpose; and it is also confirmed by the constant agreement of the conclu. sions derived from them, with each other, and with experience.
When light is reflected or refracted at a polished surface, the motion of the general body of the rays is conformable to the laws above laid down: some are, indeed, thrown to the eye in whatever situation it is placed; anil, consequently, a part of the light is dispersed, in all directions, by the irregularity of the medium upon which it is incident. This dispersion is bowever, much less than would necessarily be produced, were the rays reflected or refracted by the solid parts of bodies. II. On the Reflection of Rays at Plane and Spherical Surfaces.
Definitions.-1. By a pencil of rays, we understand a number of rays taken collectively, and distinct from the rest.
These pencils consist either of parallel, converging, or diverging rays.
Converging rays are such as approach to each other in their progress, and, if not intercepted, at length meet.