CATOPTRICS
, the science of reflex vision; or that part of opties which explains the laws and properties of light reflected srom mirrors, or specula.
The first treatise extant on catoptrics, is that which was compofed by Euclid: this was published in Latin in 1604 by John Pena; it is also contained in Herigon's Course of Mathematics, and in Gregory's edition of the works of Euclid: though it is suspected by some that this piece was not the work of that great geometrician, notwithstanding that it is ascribed to him by Proclus in lib. 2, and by Marinus in his Preface to Euclid's Data. Alhazen, an Arabian author, composed a large volume of optics about the year 1100, in which he treats pretty fully of catoptrics: and after him Vitello, a Polish writer, composed another about the year 1270. Tacquet, in his Optics, has very well demonstrated the chief propositions of plane and spherical speculums. And the same is very ably done by Dr. Barrow in his Optical Lectures. There are also Trabe's Catoptrics; David Gregory's Elements of Catoptrics; Wolfius's Elements of Catoptrics; and those of Dr. Smith, contained in his learned and very elaborate Treatise on Optics; and many others of less note.
As this subject is treated under the general term Optics, the less need be said of it here. The whole doctrine of Catoptrics depends upon this simple principle, that the angle of incidence is equal to the angle of reflection, that is, that the angle in which a ray of light falls upon any surface, called the angle of incidence, | is equal to the angle in which it quits it when reflected from it, called the angle of reflection; though it is sometimes defined that the angles of incidence and reflection, are those which the incident and reflected rays make, not with the reflecting surface itself, but with a perpendicular to that surface, at the point of contact, which are the complements to the others: but it matters not by what name these angles are called, as to the truth and principles of the science; since, if the angles are equal, their complements are also equal. This principle of the equality between the angles of incidence and reflection, is mere matter of experience, being a phenomenon that has always been observed to take place, in every case that has fallen under observation, as near at least as mechanical measurements can ascertain; and hence it is inferred that it is a universal law of nature, and to be considered as matter of fact in all cases. Thus, let AC be an incident ray falling upon the reflecting surface DE, and CB the reflected ray, also CO perpendicular to DE; then is the angle ACD = BCE, or the angle ACO = BCO.
Of this law in nature, viz, the equality between the angles of incidence and reflection, it is remarkable, that in this way, the length or rout AC + CB, in a ray passing from any point A to another given point B, by being reflected from any surface DE, is the shortest possible, namely AC + CB is shorter than the sum AG + GB of any other two lines inflected at the line DE; and hence also the passage of the ray from A to B is performed in a shorter time than if it had passed by any other way.
From this simple principle, and the common properties of lineal geometry, the chief phenomena of catoptrics are easily deduced, and are as here following.
1. Rays of light reflected from a plane surface, have the same inclination to each other after reflection as they had before it. Thus, the rays AC, AI, AK, issuing from the radiant point A, and reflected by the surface DE into the lines CF, IL, KM; these latter lines will have the same inclination to each other as the former AC, AI, AK have. For draw ABG perpendicular to DE, and produce FC, LI, MK backwards to meet this perpendicular, so shall they all meet in the same point G, and AB will in every case be equal to BD: for the incident [angle] ACB is equal to the re- flected [angle] FCE, which is equal to the opposite angle BCG; so that the two triangles ABC, GBC, have the angles at C equal, as also the right angles at B equal, and consequently the 3d angles at A and G equal; and having also the side BC common, they are equal in all respects, and so AB = BG. And the same for the other rays. Consequently the angles BGC, BGI, BGK are respectively equal to the angles BAC, BAI, BAK; that is, the reflected rays have the same inclinations as the incident ones have.
2. Hence it is that the image of an object, seen by reflection from a plain mirror, seems to proceed from a place G as far beyond, or on the other side of the reflecting plane DE, as the object A itself is before the plane. This is when the incident rays diverge from some point as A.
But if the case be reversed, and FC, LI, MK be considered as incident rays, issuing from points F, L, M, and converging to some point G beyond the reflecting plane; then CA, IA, KA will become the reflected rays, and they will converge to the point A as far before the plane, as the point G is beyond it.
So that universally, when the incident rays diverge from a point A, the reflected rays will also diverge from a point G; and when the incident rays converge towards a point G, the reflected ones will also converge to a point A; and in both cases these two points are at equal distances on the opposite sides of the reflecting plane DE.
3. Parallel rays reflected from a concave spherical surface, converge after reflection. For, let AF, CD, EB be three parallel rays falling upon the concave surface FB, whose centre is C. To the centre draw the perpendiculars FC and BC; also draw FM making the reflected angle CFM equal to the incident angle CFA; and in like manner BM to make the angle CBM = the angle CBE; so shall the rays AF and EB be reflected into the converging rays FM and BM. As to the ray CD, being perpendicular to the surface, it is reflected back again in the same line DC.
4. Converging rays falling upon the concave surface are made to converge more. Thus, let GB and HF be the incident rays: then because the incident angle HFC is larger than the angle AFC, therefore the equal | reflected angle NFC is greater than the reflected angle MFC, and so the point N is below the point M, or the line FN below the line FM; and in like manner BN is below BM; that is, the reflected rays FN and BN are more converging in this case, than FM and BM in the other.
5. The focus to which all parallel rays, falling near the vertex D, are reflected, is in the middle of the radius M. For, because the [angle] MFC = [angle] AFC which is = the alternate [angle] FCM, therefore the sides opposite these angles are also equal, namely the side FM = CM; consequently when the point F is very near the vertex D, then the sum CM + MF is nearly = CD, and so CM nearly = MD, or the focus of the parallel rays is nearly in the middle of the radius.—But the focus of other reflected rays is either above or below that of the parallel rays; namely, below when the incident rays are converging, and above when they are diverging; as is evident by inspection; thus, N the reflected focus of the converging rays GB and HF, is below M; I that of the diverging rays YB and YF, is above M.
6. Incident and reflected rays are reciprocal, or so that if the reflected rays be returned back, or considered as incident ones, they will be reflected back into what were before their incident rays. And hence it follows that diverging rays, after reflection from a concave spherical surface, become either parallel or less diverging than before. Thus the incident rays MF and MB are reflected into the parallel rays FA and BE, and the rays NF and NB are reflected into FH and BG, which are less diverging; also the rays IF and IB are reflected into FK and BL, which converge.—And hence all the phenomena of concave mirrors will be evident.
7. Rays reflected from a convex speculum, become quite contrary to those reflected from a concave one; so that the parallel rays become diverging, and the diverging rays become still more diverging; also converging rays will become either diverging, or parallel, or else less converging. Thus BDF being a spherical surface, whose centre is C, produce the radii CBV and CFT which are perpendicular to the surface; then it is evident that the parallel rays AF and EB will be reflected into the diverging ones FK and BL; and the diverging rays YB and YF become BO and FP which are more diverging; also the converging rays HF and GB become FR and BS which diverge, or else KF and LB become FA and BE which are parallel, or else lastly PF and OB become FY and BY which are converging.
8. Hence, as in the concave speculum, so also in the convex one, of parallel incident rays AF and EB, the imaginary focus M of their reflected rays FL and BK, is in the middle of the radius when the speculum is a small segment of a sphere: but the reflected imaginary focus of other rays is either above or below the middle point M, viz N being that of the converging rays GB and HF, below M; but I, that of the diverging rays YB and YF, above M.
9. When the speculum is the small segment of a sphere, either convex or concave, and the incident rays either converging or diverging, the distances of the foci, or points of concurrence, of the incident rays, and of the reslected rays, from the vertex of the speculum, are directly proportional to the distances of the same from the centre of it; that is YD : ID :: YC : IC, and QD : ND :: QC : NC. For because the radius CF, or the same produced, bisects the angle YFI in the concave speculum, or the external angle YFP in the convex one, therefore YF : IF :: YC : IC; but when F is very near to D, then YF and IF become nearly YD and ID; consequently YD : ID :: YC : IC.
In like manner, because CF bisects the angle QFN in the convex, or its external angle NFH in the concave speculum, therefore QF : FN :: QC : NC; but when F is very near to D, then QF and FN become nearly QD and ND; and therefore QD : ND :: QC : NC.
For example, suppose it were required to sind the focal distance of diverging rays incident upon a convex surface, the radius of the sphere being 5 inches, and the distance of the radiant point from the surface 20 inches. Here then are given YD = 20, and CD = 5, to find ID : then the theorem YD : ID :: YC : IC, in numbers is 20 : ID :: 25 : 5 - ID, or by permutation 20 : 25 :: ID : 5 - ID, and by composition 45 : 20 :: 5 : ID = 100/45=<*>0/9= 2 2/9 the focal distance sought.
And if it should happen in any case that the value of ID in the calculation should come out a negative quantity, the focal distance must then be taken on the contrary side of the surface.
From the foregoing principles may be deduced and collected the following practical maxims, for plane and spherical mirrors, viz,
(1). The image will appear as far behind the mirror, as the object is before it.
(2). The image will appear of the same size, and in the same position as the object.
(3). Any plain mirror will reflect the image of an object of twice its own length and breadth.
(1). The image will always appear behind the mirror, or within the sphere.
(2). The image will be in the same position, but less than the object.
(3). The image will be curved, but not spherical, like the mirror.
(4). Parallel rays falling on this mirror, will have | the image at half the distance of the centre from the mirror.
(5). In converging rays, the distance of the object must be equal to half the distance of the centre, to make the image appear behind the mirror.
(6). Diverging rays will have their image at less than half the distance of the centre.
(1). Parallel rays have their focus, or the image, at half the distance of the centre.
(2). In the centre of the sphere the image appears of the same dimensions as the object.
(3). Converging rays form an image before the mirror.
(4). In diverging rays, if the object be at less than half the distance of the centre, the image will be behind the mirror, erect, curved, and magnified; but if the distance of the object be greater, the image will be before the mirror, inverted and diminished.
(5). The solar rays, being parallel, will be collected in a focus at half the distance of its centre, where their heat will be augmented in proportion as the surface of the mirror exceeds that of the focal spot.
(6). If a luminous body be placed in the focus of a concave mirror, its rays, being reflected in parallel lines, will strongly enlighten a space of the same dimensions with the mirror, at a great distance. If the luminous object be placed nearer than the focus, its rays will diverge, and so enlighten a larger space, but not so strongly. And upon this principle it is that reverberators are constructed.
Catoptric Dial, a dial that exhibits objects by neflected rays. See Reflecting Dial.
Catoptric Telescope, a telescope that exhibits objects by reflection. See Reflecting Telescope.
Catoptric Cistula, a machine, or apparatus, by which small bodies are represented extremely large, and near ones extremely wide, and diffused through a vast space; with other very pleasing phenomena, by means of mirrors, disposed by the laws of catoptrics, in the concavity of a kind of chest.
There are various kinds of these machines, accommodated to the various intentions of the artificer: some multiply the objects, some magnify, some deform them, &c. The structure of one or two of them will suffice to shew how many more may be made.
To make a Catoptric Cistula to represent several difserent scenes of objects, when viewed by different holes.
Provide a polygonal cistula, or box, like the multangular prism ABCDEF, and divide its cavity by diagonal planes AD, BE, CF, intersecting in the centre, into as many triangular cells as the chest has sides. Line those diagonal partitions with plain mirrors; and in the sides of the box make round holes, through which the eye may peep within the cells of it. These holes are to be covered with plain glasses, ground within-side, but not polished, to prevent the objects in the cells from appearing too distinctly. In each cell are to be placed the different objects whose images are to be exhibited; then covering up the top of the chest with a thin transparent membrane, or parchment, to admit the light, the machine is complete.
For, from the laws of reflection, it follows, that the images of objects, placed within the angles of mirrors, are multiplied, and appear some more remote than others; by which the objects in one cell will appear to take up more room than is contained in the whole box. Therefore by looking through one hole only, the objects in one cell will be seen, but those multiplied, and diffused through a space much larger than the whole box. Thus every hole will afford a new scene; and according to the different angles the mirrors make with each other, the representations will be different: if they be at an angle greater than a right one, the images will be monstrous, &c.
To make a Catoptric Cistula to represent the objects within it prodigiously multiplied, and diffused through a vast space.
Make a polygonous cistula or box, as before, but without dividing the inner cavity into any apartments, or cells; line the insides CBHI, BHLA, ALMF, &c, with plane mirrors, and at the holes pare off the tin and quicksilver, to look through; place any object in the bottom MI, as a bird in a cage, &c.
Now by looking through the aperture hi, each object placed at the bottom will be seen vastly multiplied, and the images removed at equal distances from one another, like a great multitude of birds, or a large aviary.
