Treatise on Light

Refraction and

y be desired. For though I do not see yet that there are means of making use of these figures, so far as relates to Refraction, not only because of the difficulty of shaping the glasses of Telescopes with the requisite exactitude according to these figures, but also because there exists in refraction itself a property which hinders the perfect concurrence of the rays, as Mr. Newton ha

nt D in the straight line AB. I say that, whether by reflexion or by refraction, it is only necessary to make this surface such that the path of the light from the point A to all points of the curved line CDE, and from thes

of 3 to 2 (which is the same, as we have shown, as the ratio of the Sines in the refraction), it is only necessary to make DH equal to 3/2 of DB; and having after that described from the centre A some arc FC, cutting DB at F, then describe another from centre B with its semi-diameter BX eq

e the whole line AH will represent the time along AD, DB. Similarly the line AC or AF will represent the time along AC; and FH being by construction equal to 3/2 of CB, it will represent the time along CB in the medium; and in consequence the whole line AH will represent also the time along AC,

tend towards B, let there be supposed a point K in the curve, farther from D than C is, but such that the straight line AK falls from outside upon the curve which serves fo

ference at right angles. Similarly, having taken any other point L in the curve, one can show that in the same time as the light passes along AL it will also have come along AL and in addition will have made a partial wave, from the centre L, which will touch the same circumference KS. And so with all other points of the curve CDE. Then at the moment that the light reaches K the arc KRS will be the termination of the movement, which has spread from A through DCK. And thus this same arc will constitute in the medium the propagation of the wave emanating from A; w

from the centre A with its semi-diameter AF equal to 3/2 of GX; or rather, having described, as before, the arc CX, it is only necessary to make DF equal to 3/2 of DX, and from-the centre A to strike the arc FC; for these two construct

e were some material of a mirror of such a nature that by its means the force of the rays (or, as we should say, the velocity of the light, which he could not say, since he held that the movement of light was i

FC, which previously was an arc of a circle, is here a straight line, perpendicular to DB. For the wave of light DN, being likewise represented by a straight line, it will be seen that all the points of this wave, travelling as far as the surface KD along lines parallel to DB, will advance subsequently towards the point B, and will arrive there at the sa

; FB will be a - y; CB will be sqrt(xx + aa - 2ay + yy). But the nature of the curve is such that 2/3 of TC together with CB is equal to DB, as was stated in the last construction: then the equation will be between (2/3)y + sqrt(xx + aa - 2ay + yy) and a; which being reduced, gives (6/5)ay - yy equal to (9/5)xx; that is to say that having made

ines perpendicular to BA, because they represent arcs of circles the centre of which is infinitely distant. And the intersection of the perpendicular CX with the arc FC will give the point C, one of those through which the curve ought to pass. And this operates so that all the parts of the wave of light DN, coming to meet the surface KDE, will advance thence along parallels t

at of the Ovals the first of which we have already set forth. The second oval is that which serves for rays that tend to a given point; in which oval, if the apex of the surface which receives the rays is D, it will happen

worthy of remark that in one case this oval becomes a perfect circle, namely when the ratio of AD to DB is the same as the ratio of the refra

from the point A, shall deviate them toward the point B. Then considering this other curve as already known, and that its apex D is in the straight line AB, let us divide it up into an infinitude of small pieces by the points G, C, F; and having drawn from each of these points, straight lines towards A to represent the incident rays, and other

ken as the radius of the circle. Then LK ought to be to GQ as 3 to 2; and in the same ratio MG to CR, NC to FS, OF to DT. Then also the sum of all the antecedents to all the consequents would be as 3 to 2. Now by prolonging the arc DO until it meets AK at X, KX is the sum of the antecedents. And by prolonging the arc KQ till it meets AD at Y, the sum of the consequents is DY. Then KX ought to be to DY as 3 to 2. Whence it would appear that the curve KDE was of such a nature that having drawn from some point which had been

a figure not only plane or spherical, or made by one of the conic sections (which is the restriction with which Des Cartes proposed this problem, leaving the solution to those who should

ss receives rays coming from the point L. Furthermore, let the thickness AB of the middle of the glass be given, and the poin

om the point L to the surface AK, and from thence to the surface BDK, and from thence to the point F, shall be traversed everywhere in equal tim

the straight line AL; which, as is clear, make up a given length. Or rather, by deducting from each the length of LG, which is also given, it will merely be needful to adjust FD up to the straight line VG in such a way that FD together with 3/2 of DG is equal to a given straig

ce of wave which was at D will have spread into the air its partial wave, the semi-diameter of which, DC (supposing this wave to cut the line DF at C), will be 3/2 of EB, since the velocity of light outside the medium is to that inside as 3 to 2. Now it is easy to show that this wave will touch the arc BP at this point C. For since, by construction, FD + 3/2 DG + GL are equal to FB + 3/2 BA + AL; on deducting the equals LH, LA, there will remain FD + 3/2 DG + GH equal to FB + 3/2 BA. And, again, deducting from one side GH, and from the other side 3/2 of AS, which are equal, there will remain FD with 3/2 DG equal to FB with 3/2 of BS. But 3/2 of DG are equal to 3/2 of ES; then FD is equal to FB with 3/2 of BE. But DC was equal to 3/2 of EB; then deducting these equal lengths from one side and from the other, there will rema

also given in the axis the point L and the thickness BA of the glass; and let it be required to find the other surface KDB, which receiving rays that

BC must arrive at the same time at the point L; or rather all the parts of a wave emanating from the point L must arrive at the same time at the straight line BC. And for that, it is necessary to find in the line VGD the point D such that having drawn DC parallel to AB, the sum of CD, plus 3/2 of DG, plus GL may be equal to 3/2 of AB, plus AL: or rather, on deducting from both sides GL, which is given, CD plus 3/2 of DG must be equal to a given length; which is a still easier problem than t

be AB. Also let the point L be given in the axis behind the glass; and let it be supposed that the rays which fall on the surface AK tend to this point,

ught to pass. Let us suppose that it has been found: and about L as centre let there be described GT, the arc of a circle cutting the straight line AB at T, in case the distance LG is greater than LA; for otherwise the arc AH must be described about the same centre, cuttin

ather, having taken AS equal to 2/3 of AT, I observe that 3/2 of GD ought to be equal to 3/2 of SB, plus BQ; and, deducting both of them from FD or FQ, that FD less 3/2 of GD ought to be equal to FB less 3/2 of SB. And this last difference is a given length: and

iece G of the wave GT arrives at D at the same time that the piece T arrives at Q, which is easily deduced from the construction, it will be evident as a consequence that the partial wave generated at the point D will

ble thing touching the uncoordinated refraction of spherical, plane, and other surfaces: an effect which if ignored might cause some doubt conc

sparent body, which are cut at right angles by the converging rays? For they can not be spherical. And what will these waves become after the said rays begin to intersect one another? It will be seen in the solution of this

d will have also formed spherical partial waves of which these points are the centres. And the surface EK which all those waves will touch, will be the continuation of the wave AD in the sphere at the moment when the piece D has reached E. Now the line EK is not an arc of a circle, but is a curved line formed as the evolute of another curve ENC,

we may assume the small piece FP as a straight line perpendicular to the ray GM, and similarly the arc GF as a straight line. But GM being the refraction of the ray RG, and FP being perpendicular to it, QF must be to GP as 3 to 2, that is to say in the proportion of the refraction; as was shown above in explaining the discovery of Des Cartes. And the same thing occurs in all the small arcs GH, HA, etc., namely that in the quadrilaterals which enclose them the side parallel to the axis is to the opposite side as 3 to 2. Then also as 3 to 2 will the sum of the one set be to the sum of the other; that is to say, TF to AS, and DE to AK, and BE to SK or DV, supposing V to be the inters

me curve in the opposite sense. Thus the wave KE, while advancing toward the meeting place becomes abc, whereof the part ab is made by the evolute bC, a portion of the curve ENC, while the end C remains attached; and the part bc by the evolute of the portion bE wh

falls upon the refraction of the ray DE, which I now suppose to touch the sphere. The folding of the waves of light begins from the

another purpose. And it is to be noted that a straight line equal in length to this curve can be given. For since it together with the line NE is equal to the li

me. All the reflexions of those rays which fall upon the quarter-circle AB will touch a curved line AFE, of which line the end E is at the focus of the hemisphere, that is to say, at the point which divides the semi-diameter BD into two equal parts. The points

hen the piece G shall have reached I, it will be the curves HF, FI, generated as evolutes of the curves FA, FE, both beginning at F, which together constitute the propagation of the part AG. And a little afterwards, when the part AK has met the surface AM, the piece K having come to M, then the curves LN, NM, will together constitute the propagation of that part. And thus this folded wave will continue to advance until the point N has rea

ay as the mensuration of the preceding curve; though it may also be demonstrated in other ways, which I omit as outside the subject. The area AO

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