Sudden Remarks Concerning Leibniz’s Universal Principle of Optics, as to its use in Mirror Optics
SUDDEN REMARKS CONCERNING LEIBNIZ’s UNIVERSAL PRINCIPLE OF OPTICS,
AS TO ITS USE IN MIRROR OPTICS
Dissertation presided by
MARTIN JOHAN WALLENIUS
Professor of Mathematics at the Royal Academy of Åbo
submitted to public assessment by
ANDERS JOHAN LEXELL
Turku / Åbo, 1759
Printed in Stockholm by Lorentz Ludvig Grefing
In recent times, when considerable growth has been witnessed in almost every science, especially those sciences have made much progress, which are concerned with the investigation of the qualities of bodies. For, as is commonly known, while almost everyone of these [sciences] are old, fruitless and crowded with the most absurd arguments, when recently delivered, they are now in the broadest way remarkable. Not only those, which according to the ancients were sound, are now most excellently finished, but also new ones and added with a manifold of the most accurate experiments, and moreover, these ingenious and equally solid experiments are well founded demonstrations, so that in our time Physics must be referred to as one of the most elegant sciences. However, in spite of such great merits of this science recently, anyone versed in physics will easily admit, that not everything has yet come to this point. Indeed there are as yet given several properties of bodies , which either lie completely hidden for us, or they become known only faintly. The limits of the human mental capacity are so strictly circumscribed, that not without great difficulties can it to some extent reach to the concealed truths. Besides that there is an immensity of natural things worthy of our attention, even if only a few are just sufficient for the human spirit to examine. Now, to pass over to the remaining part of Physics, talk about Optics; in which, of course, however excellently elaborated by the most learned of men, especially by the famous Newton, there still remains much insufficient, and this they even admit themselves. Thus we have found the scholars to be divided in various opinions concerning the question of deriving a principle by which the phenomenon or reflexion can be explained. Indeed, Descartes and his followers attribute the reflexion of light to globules, called the second elastic element, and the surface of solids reflecting the impulse of particles (a). Others derive it by means of the inflammable and oily spirits residing in reflecting bodies. We insert in sect. §1 a mention about the Newtonian explanation of reflection. Others together with de Maupertuis (b) appeal to the general principle in physics, called the least action principle. But under this so to speak general principle there is one which is adopted by several, namely the rule that light is reflected through the shortest path, and by assuming this hypothesis they arrive at the correct law of reflection. This has been especially elaborated by the illustrious Leibniz, who in the Acta Eruditorum of Leipzig in 1682. p. 185 assumed this as a universal principle for both Optics, Catoptrics and Dioptrics; that light takes the easiest path from the radiating point to the enlightened point, i.e. if it is talked about light propagating in the same medium, the shortest path. This great Man does of course not consider to admit that the obtaining of a goal is quite inappropriate. It easily assumes the opinion that the Divine wisdom might be unworthy, as they do not found on natural laws, when light is bound to take the easiest route. Yet, as is easily shown, this hypothesis about the motion of light, which is named after Leibniz posthumously, was not quite unknown before His age. In fact, besides certain older ones, such as Vitelo, Ptolemy and Hero the mechanic were talking about reflected light taking the shortest path; of more recent ones, Fermat and others used this hypothesis (c), which even Snell and Leibnitz himself suspected, that similar thoughts applied to Dioptrics as well. And it is on the whole certain that in Fermat’s meaning all light, even refracted, must acquit its motion in the shortest possible time (d).
However this may be, the hypothesis is far from being suitable as a principle for optics on the whole; instead, it can be easily shown by examples, which concern the reflection of light, that it does not hold universally, as there are given not only a few cases, in which the path of light is not the shortest but the longest, or indeed something else, in which neither of these laws prevail. This thesis presents some sudden remarks in the principle of Leibniz, which regard catoptrics [=mirror optics]. I say sudden and sparsely collected, but not of such kind, which pertain to the discussion using a multitude of principles of all kinds. That indeed, pressed by a short time, is us forbidden, as it to a lesser extent would have freed us from obstructing our duty. We also unreservedly admit to be dispersing, on the occasion given by the present argument, some simple thoughts by somewhat distant principles; from which fact, that not only a few of these are given which by displaying the same things, if not doing anything at all, are not seen to be really necessary; yet again there would have been many perhaps neglected ones, which have lead to more from the facts. Notwithstanding this defect, we hope that the requested thing would come to us more easily, than just by wanting to show the falsehood of Leibnitz’ principle by some special examples. If for the rest, Benevolent Reader, you judge some of our work as a favour, then we have something to which we can congratulate ourselves.
(a) Principles of Philosophy P. III § 63, 64. Dioptrics Chapt. I § 8.
(b) Maupertuis Essays on cosmology p. 41
(c) Vitelo Perspective. Prop. 20. Ptolemy in Book I on reflection, Acta Erudit. Leipzig. 1682. p. 185 as well as 1701, p. 20.
(d) G.W. Kraft Lectures in Physics P. III. §§ 124, 170.
The general law, according to which all reflection of light comes about as proved by experiments, and against which all hypotheses are examined, which are used to explain this reflection, is that the angle of reflection be equal to the angle of incidence, whatever be the perpendicular on the common plane of the reflecting surface. The first of this principle of Catoptrics is consistent with the same law of reflection of elastic bodies impinging on an immovable obstacle, in agreement with the hypothesis exhibiting the shortest path, tacitly assuming that light impinges on the same reflecting surface and indeed, a rectilinear propagation. For truly the light which is reflected does not intrude into the solid parts of the reflecting bodies, for which much weighing arguments has been advanced by Newton in Opticks Bk. II P. III. Prop. 8, who also derived in the Mathematical Principles of Natural Philosophy P. I. prop. 96 and its Scholion the most likely cause of reflection (as well as refraction, cf. also Opticks Bk. I. P. I. prop. 6) being the repelling or attracting force, by which the body at the shortest distance acts on the rays of light in lines perpendicular to the surfaces and certainly equally at equal distances; from which it would also follow that the rays incident on a body and moving away from it are bent. We may nevertheless concede this bending even by denying the impact, in the present work, where the cause of the reflection is not a question, as having no reason, and rather let the rays of light be moving everywhere in straight lines and arrive at the reflecting surface. Which admitting, the firm laws on reflection, approved by every physicist, which we mentioned above, must determine, whether the path of light always be a minimum. We will also show that the path of light incident on a plane or concave surface is indeed always a minimum, but if the question is about hollow mirrors, it does not apply universally; rather the path is sometimes a maximum, also it may be neither a maximum or a minimum.
Having thus encountered several cases, in which Leibniz’s principle either does take place or does not, it is at first appropriate to consider the path of light falling upon plane as well as on convex surfaces. To this end we put forward the following proposition. (Fig. 1). Determine on a plane or a convex surface a point C located such that the sum of the straight lines AC CB from given two points A and B placed outside the same and drawn to C be the smallest possible. The first thing which is evident is that the point C must be situated in a plane which passes through A and B, if the given surface be a plane, precisely in the common section of that plane and this surface.
For if the surface be a plane, and in it a certain other point K is taken outside the section, which now will be a straight line and be named DE, taken at this normal KC, and the junction KA KB CA CB, KC will be perpendicular to the plane ACB and therefore also to the straight lines CA CB (Euclid Elements Bk XI Defin. IV. III). In the triangles ACK BCK, which are rectangular, AK > AC and BK > BC (Euclid Elements Bk I. 19 or 47), hence AK + BK > AC + CB. From which it follows that this point is not to be searched for outside the section DE, but in the very section itself. If the surface be convex, the same rule could be demonstrated here. Thus let this point be either a straight line DE, or a certain convex curve GCc pointing towards the points A B. Let there in the plane ABDE and, hence, on it, any perpendicular surface CF be drawn, and let AF BH be perpendicular to that CF. Let there be another point c on the same section infinitely near C, whereby Cc is always regarded as straight. Furthermore, with Ac Bc drawn from the centres of the rays A and B, AC and BC would describe the arcs Cn, Cr, which are normal to AC, Ac and CB, and owing to their smallness, they can be taken as straight. Then because the angle BHC = Crc, both of which are right, and rCc = HBC, for HCB is from both sides complemented by a right angle; then the triangle HBC ~ rCc and hence BH : BC :: Cr : Cc. In the same way it is shown that the triangle AFC ~ Cnc. (If you wish, these certainly follow easily from MacLaurin’s Traitise of Fluxions § 206.) Now since the fluent quantity AC + CB is a minimum, their fluxions [differentials] AC and CB must be equal, that is, cn = Cr, or put in another way, AF : AC :: BH : BC, and even (Euclid Bk. VI. 7) the angles ACF = BCH, and therefore ACD = BCE as well. It is therefore required that the point C is simultaneously so placed, that the straight lines AC, BC make equal angles with the underlying surface at C, or with the perpendicular (e).
Now, if this surface would be a plane and hence the section a straight line DE, the point C can be easily determined if, using to the given construction, from either of the given points as from B there be produced a normal (which falls in the section making a right angle with it; Euclid XI. 38 and Def. 3) BE to the underlying plane, so that EL = BE, and joining AL, which indeed occurs with the plane in the point C searched for. However, we are lead to this construction when investigating any of the values of AF = DC = x or EC = HB = a – x (if DE was considered, now given as = a). Here of course are given any AD, BE, which are named b, c, respectively. And by the demonstration the triangle AFC ~ BHC or the triangle ADC ~ BEC, then b : x :: c : a – x and therefore b + c : b :: a : x, and so b + c : c :: a : a – x. Consequently, x or a – x are easily found by this construction. If indeed we would draw LN parallel to DE, which meets the continuation of AD at N, then EL = DN and so AN (b + c) : AD (b) :: NL or DE (a) : DC (x), which determines the point C itself.
As we have indeed found analytically, the sum AC + CB is the smallest, if the angle ACF = BCH or ACD = BCE, or again; it can likewise be most easily proved from the primes of Geometry; of course, if the surface was Iº a plane, in the same way as is seen everywhere in e.g. Musschenbroeck’s Elements of Physics § 1036, Instit. Phys. § 1237, Kraft’s Lectures in Physics P. III. § 124, of course if it be added to it, which has been adduced above, that the point C is not to be searched for outside the section. But if the surface would be IIº convex, it is proved in the following manner: By taking on the convex surface GCc any point I apart from C, and by joining IA and IB, I maintain that AC + CB < AI + IB. Because the considered surface is convex, the point I can never be located on the side of AB of the plane DE touching it in C, but on the opposite side or even on the plane itself (f). If so, then AC + CB < AI + IB by Case I. But if not so, AI + IB will transgress this plane at any of its points. If O be the point where AI crosses this plane, and joining BO; then by virtue of Case I, AC + CB < AO + OB and (Euclid Elem. I. 20) AI + IB > AO + OB, and therefore a fortiori AI + IB > AC + CB.
(e) Even when the curve GCc is concave, directed away from the points A B, the above analysis is nonetheless valid; in that case, however, the point C is so defined, that AC + CB is not always determined to be a minimum, but also a maximum, as will be shown in the sequel. Indeed this method is in itself equally apt for determining a minimum as well as a maximum; which case will take place in a special case must be judged from the innate nature of the case.
(f) If the surface would be a cylindrical or a conical section, it is showed later that there are indeed points of a common tangent plane .
On account of the preceding paragraph, if A B are two points, one of which emits light which comes to the other point through reflection in a plane or a convex mirror; by supposing the shortest path for light leads to equality of the angles of incidence and reflection, or again from this equality follows the shortest path. And just as the hypothesis of Leibniz is always valid when it is talked about light which suffers a reflexion in one plane or convex mirror, so it will be equally valid, if the reflection is repeated in several such mirrors. (Fig. 2). Thus introducing another different plane outside of the mirror will produce two points [of reflection]. Therefore the ray of light landing from either of these point must first be reflected in one mirror and then arrive at the other. The points of incidence and reflection of the rays of light are to be determined. Let first FG, GH be two mirrors, and the points A, b, one radiating light and the other illuminated. If the ray emitted by A should first impinge on FG, then send out from A a perpendicular AF prolonged to the other side such that AF = Fa. In the same way, from the point b send out a normal to GH (or to its plane, if needed continuing, which saying applies in any other case) prolonged until Kβ be equal to bk. Now draw the straight line aβ; this crosses the mirrors in the points searched for C, D, and joining AC, bD we have the path of the ray ACDb = aβ (g). Indeed when the angle AFC = aFC and aF = AF, then (Euclid I. 4. 15.) the angle ACF = (aCF =) DCG; and similarly the angle bDK = CDG, that is the angle of reflection is on both sides equal to that of incidence. Furthermore, because aC = AC, bD = βD, then AC + CD + Db = (aC + CD + Dβ =) aβ. Now let there be three mirrors FG GH HI, while the points are given at A B. Determine as for the previous point a, sending from B to the mirror HI, in which the last reflection happens, the perpendicular reflection prolonged to b so that IB = Ib, and again from b to GH a perpendicular prolonged to β such that Kβ = Kb. Then draw aβ, which gives the desired points C and D, and joining further Db determines the point E searched for in the mirror HI. And so drawing AC, BE, the path of the ray ACDEB = aβ, which is also shown above by similar reasoning.
But if there were still more mirrors, several similar operations should have to be employed, similar to those made above, as is easily understood from the said. Also it seems to have the same solution, if the points A, B (or b) coincide, that is, if light from A arrives at the same point from which it departs, in which case the point must recede backwards. But in all these cases, the path of light is always a minimum. For if the rays of light would impinge on any other points of the mirrors as c d, while going e.g. from A to b, this path would be Acdb > ACDb. Indeed, we have (Euclid Elem. I. 20.) ac + cd + dβ > aβ, because Ac = ac and bd = βd, Ac + cd + db > AC + CD + Db. The same applies if we consider the path of light, as ACDEB, reflected by several mirrors.
(g) Here it is supposed that the line aβ crosses the mirrors, FG, GH. Unless that would have occurred, it would indeed be impossible for light to go from A to b by such kind of reflection, as has been supposed.
From examining the reflection in plane and convex mirrors, we now proceed to the reflection from hollow (i.e. concave) mirrors. And indeed first, in order to be able to pronounce on the path of light reflected from hollow spherical mirrors, it seems fit to put forward another proposition in the form of a lemma. In particular, it is this: On the diameter of a circle, or the same prolonged, let two points be chosen no matter how, their distance being equal to the centre, the square of the straight lines from these points taken to any point on the periphery, always produce a constant sum (h) namely equal to the square of the semi-diameters and the distance of either these points to the centre taken two times, or truly the sum of the square of the diameters, into which it is divided by either of these points. Let EIF be a circle in whose diameter EF there be taken the points A B equally distant from the centre C, from which two straight lines GA, GB are drawn to any point G on the periphery. Let GC be drawn and send out from G to the diameter a perpendicular GD. Indeed, because AG2 = AC2 + CG2 + 2·AC · CD and GB2 = GC2 + CB2 – 2·CB · CD (Euclid Elem. II 12. 13.); then thanks to AC = CB, the sum AG2 + GB2 = 2·CG2 + 2·AC2 is exactly constant, clearly given for the circle and the points A, B. And further, because CG = CF, CG2 = CF2 from which AG2 + GB2 = 2·CF2 + 2·AC2; but (by Euclid II. 10. or 9.) AF2 + BF2 or AF2 + AE2 = 2·AC2 + 2·CF2; therefore AG2 + GB2 = AF2 + AE2 (i).
(h) That the curve, whose properties are these, is a circle, is easily proved by changing a little the series of demonstrations applied in this paragraph. The proposition is of course: from the given points A and B, to find the curve EGF, from any point G of which are drawn the straight lines to A and B, the squares of which summed is a constant. Let AB be joined, on which, prolonged if needed, a perpendicular GD will be sent out; let AB be bisected at C and CG be joined. Presuming these, it will be demonstrated as in this §, that GA2 + GB2 = 2·CG2 + 2·CA2. Just as from the hypothesis that GAq + GBq be constant, also 2·CGq + 2·CAq or its half CGq + CAq will be constant. Whereby as CA be given and thereby CA2, the remaining CG2 is some constant, consequently the straight line CG from a given point C drawn at any point of the curve, has a constant magnitude. From which is seen that the curve searched for is a circle whose centre is C while the radius is CG. If it pleases to use calculus, let AB be a, CD x, GD y; then AD = ½ a + x, BD = ½ a – x or x – ½ a; and let AG2 + BG2 = the given square bb. Thus AGq = (ADq + GDq =) ¼ aa – ax + xx +yy, consequently AGq + BGq or bb = ½ aa + 2 xx + 2 yy from which yy + xx = ½ bb – ¼ aa, which is the equation of a circle. Thus the curve searched for is a circle centred at C, describing in this way a radius = √(½ bb – ¼ aa). From A prolonged along the line CA draw AK = ½ b which is equal and perpendicular to KH. Then from the centre A the interval AH would describe an arc cutting in I a perpendicular erected from C. If now from the centre C the radius CI would describe a circle, then it will satisfy the problem.
(i) The present proposition stands as the most simple and most special case, contained within the general and very remarkable properties of the sphere, demonstrated in the Analyse des infiniment petits § 33 by the Marquis de l’Hôpital.
The other preliminary proposition is this (Fig. 4): Among the countless triangles ABC established on the same base AB and having a given angle ACB opposite to the base, the one is searched for, having the sum of the legs AC + BC a maximum. Let BC be prolonged towards D until CD = CA, consequently AC + CB = BD, and and let AD be joined. Then (Euclid I. 5. and 32.) the angle ACB will be twice the angle D. Therefore as the angle ACB is given, also D is given. And so by (Euclid III. 21. and earlier) all points D assembled on the circumference of the circle ADEB, whose segment above the straight line AB is understood to constitute the given angle, certainly equal the half of the given ACB. The longest of all chords BD in this circle (Euclid III. 15.) is the diameter. But when BD is the same as the diameter the point C on it cannot be but (Euclid III. 7.) the centre of the circle, thanks to CA, CD being equal (from the construction). Thus in this case, where evidently BD or AC + CB is a maximum, CA = CB, that is of all the triangles having the same base and the same opposite angle, the isosceles triangle has the greatest sum of the legs.
This fact can also be easily determined by the method of maximum and minimum (k) by means of the calculus of fluxions. Let the base AB be constant = b, AC = x – CB = y. From one of the legs BC, prolonged if needed, let us conceive drawn a perpendicular AF from the vertex of the opposite angle. Thanks to C being constant, the ratio between AC and CF (Euclid VI. 4.) which is set to = 1:n, hence CF = nx. Now, it is true that (Euclid II. 12. or 13.) AB = AC2 + CB2 ± 2· CB · CF, i.e. xx + yy ± 2 nxy = bb. Hence taking the fluxions [=differentiating] and dividing all terms by 2 gives x dx + y dy ± ny dx ± nx dy = 0. Now as the sum AC + CB is a maximum, its fluxion is dx + dy = 0. Thus by substituting dy for – dx and dividing by dx gives x ± nx = y ± ny, and dividing by 1 ± n, x=y. Thus, in this way constructed one can easily see that the triangle searched for is an Isosceles triangle. Over the given base AB a segment of a circle would be described, which occupies the given equal angle and the arc of that segment, which is the locus of the vertices of the infinite number of triangles, by bisecting it, we have the vertex searched for (Euclid III. 33. 21. 30.). But then again it may be inferred that of all the triangles having the same sum of the legs, and having equal angles, the one with equal legs as a base stays a minimum, (l) which even can discovered by the new calculus. Let this sum be a, the leg AC = x, BC = a – x, CF as before = nx, and AC2 + CB2 ± 2· CB · CF = AB2 that is aa – 2ax + 2xx ± 2nax – (± 2nxx) = AB2. Now as AB is the smallest of all, then also its square must be a minimum; thus the fluxion = (–2a + 4x ± 2na – (± 4nx))· dx =0, whence dividing by 2 dx and transferring terms 2x – (± 2nx) = a –(± 2na) and x = (±na–a)/ (±2n–2) =½ a = a – x.
(k) In the present work it is easily understood that this method determines the case where the sum of the legs is a maximum, and not at all a minimum, but it can be decreasing until its excess over the smaller base is any assignable quantity, exactly when the triangle itself vanishes (Euclid I. 20. 22.),
(l) Again it is evident that by giving the sum of the legs, their angles included, will not give the case in which the base is a maximum, but it could be just as much as near as that sum of the legs it approaches.
Examining now some cases concerning the reflection of light from hollow spherical mirrors, we first show (Fig. 3): Taking two points A and B in the diameter EF of a hollow mirror at equal distance from the centre, then the light rays sent from one of them and reflected onto the other take the longest path, yet except for AF + FB or AE + EB, which reflections return to themselves (*), for these had been the shortest path. In any great circle of a sphere passing through EF, (of course normal to the mirror) a diameter drawn perpendicularly to EF will give in the circumference of the circle the point I of the incident light (which follows manifestly by the equality of the angles CIA CIB from Euclid I. 4.); then that path AI + IB of the incident and reflected rays will indeed be the longest. For if we understand A and B to be the foci of an Ellipse passing through I, it will totally fall outside the circle, which obviously is centred at C and has CI as its shorter semi-axis. Drawing now from A B to some point on the circle different from I, as to G, the straight lines AG, GB, one of which, such as AG, we imagine prolonged so as to meet with the ellipse in O, and joining OB, then (on account of the properties of ellipses) AI + IB (or AO + OB) > AG + GB. (Euclid I. 21.) This is overcome also as follows. Conceive the right-angled triangles, whose legs are AI IB and AG GB; their hypotenuse will be equal (Euclid I. 47.) because (§4) AIq + IBq = AGq + GBq. Thus, since AI = IB, then (see §5) AI + IB > AG + GB. And indeed the sum AG + GB becomes the smaller, the further AG and GB are from equality, i.e. (Euclid III. 7.) the closer the point G is from E or F, in which it falls on the points EF, then of course the path of light AF + FB, which is reflected bock on itself, is the smallest of all AG + GB. This proposition can also be demonstrated by the following reasoning. Drawing from the another point G on the circle, AG to L until AL makes = 2 AI or AI + IB, and bisecting AL in N so that AN = AI. Whereby on account of AG > AI (Euclid III. 7.), AG > AN; and so indeed AG would be nearer the diameter EF, the greater AG (Euclid local citation) and thus also NG. And because AL is cut in equal parts in N, but in unequal parts in G, then (Euclid II. 9.) AG2 + GL2 = 2·AN2 + 2·NG2; or owing to 2·AN2 = (2·AI2 = 2·AI2 + 2·IB2 = §4) AG2 + GB2, then AG2 + GL2 = AG2 + GB2 + 2·NG2, consequently GL2 = GB2 + 2·NG2; such that GL > GB always prevails and thus adding the common AG, AL or AI + IB > AG + GB. Moreover 2·NGq = (GLq – GLq = Euclid II. 5. cor.) GL + GB · GL – GB. Now just as the point G approaches nearer to F the extremities of the diameter EF come nearer; for then (as shown) increasing NG, AL and AN being constant, thus decreasing GL and (Euclid III. 7.) GB and hence GL + GB, then 2·NG2 increases, and this equals (as shown above) the content of the rectangle formed by GL + GB and GL – GB; therefore (thanks to decreasing GL + GB as demonstrated) GL – GB increases, such that also the excess of (AG + GL or) AI + IB itself increases over AG + GB or AG + GB is continually smaller down to AF + FB, which therefore is the minimum. It truly appears that of all rays emitted from A cutting perpendicularly the great circle of the sphere of diameter EF, incident on B by reflection, and moving by the longest path; of all points on the periphery this point is I.
(*) But this cannot take place unless these points A B fall inside the sphere.
As we have demonstrated in the previous paragraph, the path of light is the longest, when on the diameter of a hollow spherical mirror two points are taken at equal distance, one radiating and the other illuminated; this is believed to hold true even in other cases. Thus, (Fig. 5) taking two points R, S, on the surface of a hollow spherical mirror, a ray being emitted from one of these points and reaching the other point after reflexion from the mirror, will take the longest path, if the reflection taking place on both sides of the plane going through RS, in the plane where RS crosses the great circle of the sphere in a right angle, on which the centre of the sphere lies, or on the opposite, then the path of light is neither the minimum nor the maximum of all. Joining RS, upon which in K, cutting it in two parts, a normal IT is erected in the plane of the great circle of the sphere crossing RS, whereby IT is perpendicular to the surface of the sphere; but it crosses the periphery of this circle at the two points I and T, in which points reflection should happen, which is manifest by the laws of reflection, since (Euclid I. 4.) in the triangles RIK, SIK the angles RIK = SIK, and by similar reasons the angles RTK = STK, while IT is perpendicular to the diameter (Euclid III. 1. Cor.) and hence normal to the circle (Euclid III. 18.). Now taking first on the circumference of the circle, on that side of the straight line RS on which the centre of the circle or the sphere lies and obviously where [the point] I is located, some other point G, and drawing RG, SG, then RI + IS > RG + GS. When indeed the angle RIS = RGS (Euclid III. 21.) but RI equals IS, whereupon it follows (§5) that in the equilateral triangle RIS the sum of the legs RI + IS will be > RG + GS. Then suppose some other point O on the spherical surface not lying on the circumference of the circle RIS, and drawing RO SO, yet I say that RI + IS > RO + OS. For by cutting the sphere through ROS, which section in fact is a small circle of the sphere, which revolves around RS until its plane falls into the plane of the great circle RIST and the point O indeed towards the parts I of the plane given by RS; which being made it is easily understood, that this small circle, which obviously cuts the great circle at RS, its segment ROS should fall inside the circumference RIGS, while the remainder segment SNR truly should stay outside the remaining STR. Thus take any point G in RIGS such that joining RG GS, O falls inside the triangle RGS; from which it follows (Euclid I. 20.) that RO + OS < RG + GS; but (as demonstrated) RI + IS > RG + GS, therefore the much greater is RI + IS > RO + OS. Thus it is shown that RI + IS is the greatest sum of all the straight lines drawn from RS to any point on the surface of the sphere. Then, as to the other case; taking first any other point H on the circumference RTS besides T, and drawing RH HS; to this, as above (§5) applies RT + TS > RH + HS so that it is not a minimum, but a relative maximum. Yet it is not an absolute maximum. If indeed any small circle of the sphere is conceived that passes through RS and revolves around an immovable RS until this small circle falls into that very plane RIST. As already reminded above, setting up any segment at all above the straight line RS, it would fall outside the circumference of RTS. It would go as RNS; whereby the point N in RNS can be chosen such that to the drawn straight lines RN, NS, T would fall inside the triangle RNS, and thus RT + TS < RN + NS. Therefore RT + TS is neither a minimum nor an absolute maximum. (Fig. 5) But given the points P Q, from one of which P light is coming out, and after reflection from a point I arrives at the other point Q, the location of that point would be obtained either inside or outside the surface of the sphere, the part of which constitutes the mirror, so that each of these points P Q fall on the opposite sides with respect to I of the diameter EF, which crossing PIQ in the great circle of the sphere is drawn parallel to the straight line which is tangent to the point I; then, again the path of light PI + IQ is the longest. For drawing a semi-diameter IC via I, it will be perpendicular (to the tangent through I and thus clearly to a parallel to this straight line) to EF itself; wherefore, if RI IS would meet EF (prolonged if needed) diametrically in the points A B, then in the triangles ICA ICB, on account of the angles at C being right and the angles at I (by the laws of reflection) being equal, (Euclid I. 26.), AC = CB; hence taking some other point G besides I on the periphery EIF and joining AG BG, then (§6) AI + IB > AG + GB, from which by adding on both sides AP + BQ, we have PI + IQ > AP + AG + GB + BQ. It is true that (Euclid I. 20.) AP + AG > PG and GB + BQ > GQ, so that PI + IQ > PG + GQ holds even more strongly. But it could also be examined whether PI + IQ would be larger than the straight lines drawn from the points P Q to any point of the mirror chosen outside the circumference EIF taken together; certainly as it is seen to be exceedingly extensive, we refrain from the demonstration, and it suffices here to be shown that the path of light IP + PQ is not at all the shortest.
An easy demonstration of this proposition is found in Milliet Dechales in Mundus Mathematicus Vol. III. Catopt. Bk. I. p. 570.
Setting aside the hollow spherical mirror, let us now consider the simplest case of reflection from a hollow mirror formed by revolution of any conical section around an axis. And first indeed what concerns parabolic or hyperbolic mirrors is that the rays of light sent out from the focal point F of a parabolic or a hyperbolic mirror, and in turn reflected by some point I, follow the shortest path. Of course this parabolic or hyperbolic conoid is cut according to the axis AD, crossing the plane of I, where the incident ray as well as the reflected ray lie on account of §1, and of course the Apollonian parabolic or hyperbolic section, whose focus is F. First, I say, taking outside this section some point on the mirror, called v and in the mind draw Fv, vI, then the sum Fv + vI cannot be a minimum. Obviously, if Fv is thought to revolve around the axis of the curve, the point v describes the periphery of a circle, AD passing through its centre, and the plane of this circle stays perpendicular [to the axis]. After (which should happen discussing the origin of conoids) v touches the section ABG, in which Fv assumes the location of FV, joining IV, which lies in the very same plane as AD, and either converges, or is parallel, with it. This being supposed, IV is perpendicular to the plane of the circle (Euclid XI. 8.), and thus joining in the mind Vv, IVv is a right angle, whence IV < Iv. But when this is true, VI may be prolonged until the axis reaches the point P; and if Pv is thought to be joined, then PV = Pv (c.f. Wolff Elements of Geometry §487). Thus because in the triangle PIv (Euclid I. 20.) Pv + vI > PI (Fig. 6), or (Fig. 7 PI + Iv > PV) PI – PV (or PV – PI) that is IV is again < Iv, whence FV + VI < Fv + vI. Hence it appears that the sum Fv + vI cannot be a minimum. This being supposed, draw via I in the parabola the diameter IB meeting the parabola at the point B, which will be the point of incidence, while joining FB, then FBI is the path of light, just as the properties of parabolas and the laws of reflection require. I say that this path is the shortest or FB + BI < FV + VI, taking in the curve any other point V and drawing FV, VI. For drawing the diameter VE, which at E meets the straight line ID, which leads from I perpendicularly to the axis AD. Then let IB, EV be prolonged, so that they meet the directrix CG [=the defining line of any conical section]; since IG EC are parallel, as are also GC IE, IG = EC (Euclid I. 34.). And so while from the nature of parabolas it follows that FB = BG, and FV = VC, then FB + BI = (IC = EC =) FV + VE, so that every ray going from the focal point to the parabolic mirror and reflected from it to a plane perpendicular to the axis, travel an equally long path. But in the right angled triangle VEI, the side VI > VE (Euclid I. 19. or 47.) and hence FV + VI > (FV + VE =) FB + BI. As regards to hyperbolas (Fig. 7); draw from its external focal point f to the point I the straight line fI, which meets the hyperbola at B, then by the properties of hyperbolas and the laws of reflection, B is the point of incidence or FB + BI is the path of the incident and reflected rays. Let V be any another point on the curve, drawing FV VI, then FB + BI < FV + VI. Indeed draw a straight line through f V, which meets at C the circle DICG with the centre f described by the interval fI, then fB + BI = fV + VC and by the nature of hyperbolas fB – FB = (transverse to the axis =) fV – FV, and leading this to the equation before, it becomes FB + BI = FV + VC. But (Euclid III. 7.) VC < VI, therefore FV + VC or FB + BI < FV + VI. Of the given it also follows that all the rays sent out from the focal point of a hollow hyperbolical mirror and reflected by it upon a spherical surface, whose centre is the focus of the opposite hyperbola, accomplish an equal path = AD + AF, if FD be the semi-diameter of the sphere.
In fact (Fig. 8), if rays are sent out from either of the foci F of a hollow elliptic mirror reflecting the rays to the point A, the shortest of these paths is the one which comes from the same half of the longest axis on which A is located, while the longest comes from the other. It is necessary that the point of the mirror, where the rays are reflected, is located in the plane passing through the points F A and perpendicular to the elliptic mirror (§1) such as the section of an ellipsoidal surface, cut along the axis BD and A, which section BCD will of course then be an Apollonian ellipse. Draw a straight line across the point A and the other focus, which is f, this line crosses the ellipse twice, and name these two points C, c, in which (A being located inside the ellipsoid) this reflection is carried out. Then by drawing FC, Fc, we have the paths of the rays searched for FC + CA and Fc + cA, from where the nature of the ellipse and the laws of reflection follow. Now I say that the path FC + CA is the shortest but Fc + cA the longest. Of course taking any other point V either on the curve of the section or outside it and drawing VA VF Vf, then FC + CA < FV + VA. And indeed due to the properties of ellipses FV + Vf = FC + Cf but yet VA + Af > Vf (Euclid I. 20.), then FV + VA + Af > FC + Cf so that doing away on both sides the common Af, FC + CA < FV + VA, so that it is indeed the smallest that can be. But from the other side I say that Fc + cA is the greatest. For FV + Vf = Fc + cF, because added to both sides fA, FV + Vf + fA = Fc + cA. But (Euclid I. 20.) Vf + fA > VA so that Fc + cA > FV + VA. Therefore it follows that Fc + cA is the longest path. And so it is clear that the difference between the longest and the shortest paths equals 2Af, because FC + CA + Af = Fc + cA – Af (evidently FC + Cf = Fc + cf on account of the properties of ellipses) and thus FC + CA + 2Af = Fc + cA. It is moreover evident, that if a hollow mirror, as VCD were a part of the surface of a certain ellipsoid, and outside its whole extent some point a is positioned, then light from the focus F is reflected to this point a again not unless this arrives only the longest path FC + CA. Namely drawing FV, VA, Vf, then FC + Cf = FV + Vf and so by adding fa, FC + Ca (= FV + Vf + fa) > FV + Va (Euclid I. 20.).
But if on the axis of a hollow Elliptic mirror, which of course is formed by revolving an Ellipse about either of its axes (Fig. 8), two points H I be taken both at equal distance from the centre, and even located outside its foci, if the axis of the mirror was the longer axis of the Ellipse; light, which is emitted from either of these points, reaches the other by reflection passing through the longest path, except in the case, where light is reflected back on itself, if it was possible, which may be shown by similar reasoning as was done above for hollow spherical mirrors, starting around § 6. It is not impossible that what is said of spherical mirrors in §7, changing somewhat the demonstrations or adding new ones, could be applied to elliptic mirrors.
Let us conceive as depicted (Fig. 8) the foci F and f, as well as the shorter semi-axis EG of the ellipse BGD. Let GLM be some conical section having the position of the axis (transverse, indeed when it is question of a hyperbola) at GE, its peak at G, so the latus rectum [=the side line, or the chord through the focus transverse to the axis] pertaining to this axis is less than the latus rectum of the small axis of the Ellipse BGD. Because then all conical sections with the peak at the axis have the curvature of the circle, whose diameter is equal to the parameter of this axis; whence it follows that the Curve GLM, where of course the curvature of the ellipse BGD in G is smaller, as far as some part around G as GL falls inside the Ellipse. On this account FG + Gf is greater than the sum of the straight lines drawn from the points F and f to any point on the arc GL. Thus FG + Gf still does not exceed the sum of the straight lines which can be drawn from the points F and f to any point chosen on the curve GLM; unless [the point] would fall totally inside the ellipse BGD. And indeed if GLM would be a Parabola or Hyperbola, the sum of such straight lines could still be assumed to be greater, in that the distance from these curves to any given F and f may extend longer. Now if GLM was a hollow mirror, or rather a plane section through the axis of the mirror, and thus perpendicular to it; light that issues from either of the points F f, falling on the point G of the mirror, reflects back on the other [point]. But as is evident from this, the path of light FG + Gf is neither the shortest, even less the longest. So that rather, while an infinite number of curves are possible, having the axis GE and the peak G, in which the tangent is perpendicular to the axis, and which are hollow towards the axis but from which the diverging legs recede further and further away, and some part of which fall inside the ellipse BGD; as is indeed evident from the chapter at hand there are given innumerable cases in which the path of light that proceeds from F to f, but again going through a reflection in the hollow mirror, is neither a minimum nor a maximum.
Thus we have seen that the principle of the shortest path in Catoptrics, although it often holds true (§§ 2. 3. 8.) yet it evidently has many exceptions, where light often takes the longest path (§§ 6. 7. 9. 10. 11.) and indeed sometimes, unless it is the longest (§§ 6. 9. 10.), either the shortest or the longest (§§ cited); lastly there are given the following cases, in which this path is neither a minimum or an absolute maximum (§§ 7. 11.), if it is talked about the distances from points (§11) on the mirror in the very section (the plane through the radiating and illuminated point perpendicularly to the mirror), or the distance from the single universal point of reflection (§7.) to these two given points. Finally it would be a pleasure to approve this sole most easily generated example by bringing in more singular cases. Let a hollow mirror be cylindrical, formed by moving in parallel either a circle or an ellipse, or some other innate curve as we have mentioned in §11, along the straight line formed from the perpendicular, so that be it any common section of the mirror and the plane, which is parallel to the generating curve and thus perpendicular to the mirror, this curve itself is maintained equal and similar. Now taking in any such a section of a plane two points, radiating and illuminated, of which the respective location is determined in such a way as A and B in §6, H and I in §10, P and Q or R and S in §7, F and a in §9, F and f in §11. On account of the cited paragraphs it is clear that light takes the path which, while obeying the rule that the points on the mirror lie in the very same section, will be a maximum (or at least §11 not a minimum). However, by no means can it be an absolute maximum, as it appears next, because taking on this straight line, which describes the reflecting point during the mentioned motion on the surface of the mirror, some point X, its distance from the radiating and illuminated points would continually become greater, exactly as remote X is from the reflecting point. The hypotenuse is of course the greatest side of a right angled triangle; but keeping one leg unaltered, while the other is growing, also the hypotenuse is growing.
Besides to the kinds of examples given here this pertains also to hollow mirrors not formed by revolution of some curve about its axis. That this is valid not only for mirrors, conceived by rotation, in which the path of light is always an absolute maximum or a minimum, it is reminded of the example discussed in §7, where it was shown that RT + TS is altogether not a maximum, even less a minimum.
But not to ignore anything, it seems appropriate that we also throw in a mention of some refuge of all the difficulties, which the defenders of the hypothesis of Leibniz that pursue his principle perhaps have realised themselves, but try to elude. There is of course the objection which Dechales (*) himself has thrown against those wanting to oppose the hypothesis that light goes the shortest path, that a curved mirror should certainly not be considered as any single mirror, but as composed of an infinite number of plane mirrors. In this explanation also Leibniz himself wants to search protection, which the more obscure words may seem to indicate, confirming that his hypothesis applies to concave or convex surfaces, by considering their tangent planes (**). However this may be, this response is not actually seen to be of such great importance. Besides talking correctly of curved surfaces, it hardly seems proper to consider them as arising from an infinite number of planes, when one should rather talk of their origin as curves, however many their parts be, no matter how small, it is really a curve [not a plane], although on account of their smallness they can be held as plane; also granting this explanation, I for my part cannot adequately understand how that could save Leibniz’s hypothesis. Even if it would be true, that the path of light reflected from a hollow mirror is the shortest of the sum of the straight lines that can be drawn from the radiating point and the illuminated point to diverse reflecting points of the plane which are tangent to the mirror in the reflecting point; however, the same does not apply, if a comparison is made with the sum of the paths of light drawn from these points to some other curve or to a point on all the other tangent planes. Hence, supposing this explanation, it is quite impossible to apply the principle in question to hollow (concave) mirrors. As far as it is observed to planes, it is certainly valid, but not as far as any curve would be considered. But if the same principle was seen appropriate to apply to hollow mirrors, in such a way understood and enunciated: The reflection of light happens in that point of a mirror, in which, in the plane tangent to the mirror, the distance of that point to the radiating and illuminated points taken together is the smallest of these distances to any other such point tangent to the plane; we easily admit the truth of it. In what way true as it might be, the words are not so easily carved out from the citations of Leibniz, nor does there appear a reason why the same thought would come to convex mirror as to the concave, when the hypothesis of the shortest path prevails absolutely to the former [convex] (§2) but not to the latter [concave]. For the rest if it will be seen, what is discussed in these small pages, that the breakdown of the universality of the principle of Leibniz will simultaneously also weaken the principle of least action proposed by the illustrious de Maupertuis. In fact, when the quantity of action is estimated as the product of the mass of the moving body with the speed and the distance travelled, while the mass and velocity of light in the reflection stays
invariable, the principle of least action appears to coincide in this case with the principle
of the shortest path, indeed so that the doubts risen against this rule
also prevail against it.
* Mund. Mathemat. Vol III. Catopt. Bk. I. p. 570
** Acta Eridutor. Leipzig Year 1682. p. 185.
PRINCIPIVM VNIVERSAE OPTICAE LEIBNITIANVM
QVATENVS IDEM IN CATOPTRICA ADHIBETVR
CONSENSV AMPLISS. FACVLT. PHILOSOPH. IN REGIA ACADEMIA ABOENSI,
MARTINO JOHANNE WALLENIO
MATHESEOS PROFESSORE REG. ET ORD.
PVBLICAE CENSVRAE SVBJICIT
ANDREAS JOHANNES LEXELL
DIE ANNI MDCCCLIX
LOCO HORISQVE A.M. SOLITIS.
TYPIS LAVRENT. LUDOV. GREFINGII.
Recentior aetas cum ad incrementum omnium fere scientiarum plurimum contulit, tum eam inprimis promovit, quae in investigandis corporum qualitatibus occupata est. Hanc enim, quum plerique veterum sterilem & non nisi commentis absurdissimis refertam, recentioribus traderent, jam mirum fere in modum auctam esse, in vulgus notum est. Non enim ea solum, quae apud antiquiores saniora fuere, egregie jam perpolita sunt, sed etjam nova quaeque & multifaria experimenta adcuratissima addita, & quod maximum est, his experimentis ingeniosae pariter ac solidae superstructae sunt demonstrationes, adeo ut nostro aevo Physica inter elegantissimas referri debeat scientias. Tanta autem quamvis sint in hanc scientiam recentiorum merita; fatebitur tamen facile, quisquis vel aliquantulum in physicis versatus, nec hos omne tulisse punctum. Dantur enim adhuc plurimae corporum affectiones, quae nos vel latent plane, vel nonnisi obscure innotuere. Adeo scilicet arctis limitibus circumscriptum est humanum ingenium, ut non nisi magna cum difficultate ad abstrusas aliquantum veritates pertingere queat. Ea quoque vastitas est rerum naturalium, quae nostram merentur attentionem, ut paucis solummodo rimandis sufficiat industria humana. Loquitur hoc, ut reliquas Physicae partes transeamus, Optica; quippe in qua, quamvis a viris maxime eruditis im primis illustri Newtone egregie sit laboratum, multa tamen adhuc deficere, etjam hi ipsimet fateri coguntur. Ita novimus eruditos in varias abire sententias, cum quaesito oriatur de principio, ex quo phaenomena circa luminis reflexionem explicari debeant. Cartesius scilicet atque ejus asseclae; luminis reflexionem tribuunt globulorum, quos vocat secundi elementi elasticitati, atque in solidas superficiei reflectentis particulas impulsui (a). Alii eam derivant ex inflammabili & oleoso spritu in corporibus reflectentibus agente. De explicatione reflexionis Newtoniana in sec. §1 injiciemus mentionem. Cum de Maupertuis (b) alii ad generale illud in physica principium, quod actionis minimae vocari solet confugiunt. Sub hoc autem principio tanquam generali, continetur illud a nonnullis adoptatum, vi cujus lumen reflexum via ferri brevissima & ex assumta hac hypothesi reflexionis legem recte deduci posse existimant. Hanc operam egit imprimis Illustris Leibnitius quippe qui, in Actis Eruditorum Lipsiens. Anni 1682. p. 185 pro universae Opticae, Catoptricae atque Dioptricae unico principio hoc assumit; quod lumen a puncto radiante ad punctum illustrandum via perveniat facillima i.e. si de lumine per idem medium propagato sermo fuerit, via brevissima. Scilicet cogitans magnus hic Vir, non admittenda esse ea, quae fini obtinendo parum congrua videntur; facile in eam inductus est opinionem, quod sapientia Divina esset indignum, eas non condere naturae leges, ut lumen viam impenderet facillimam. Hanc de motu luminis hypothesin, quam in posterum a Leibnitio denominare, placet, non tamen ante aetatem Ipsius plane fuisse ignotam facile ostendi potest. Praeterquam enim quod quidam veterum, ut Vitellio, Ptolemaeus & Hero Mechanicus de via luminis reflexi minima locuti fuerint; ex recentioribus Fermatius aliique illam hypothesin adhibuere (c), quem quidem ut & Snellium ipse Leibnitius suspicatur similes meditationes ad Dioptricam quoque adplicuisse; & certum omnino est, ex Fermatii sententia omnem luminis etjam refracti, motum tempore brevissimo absolvi debere (d).
Quicquid autem sit, tantum abest, ut hypothesis haec pro principio quodam totius Optices assume queat; ut potius facillimis ostendi possit exemplis eam, quod ad reflexionem luminis attinet, universaliter non esse veram, sed dari casus haud paucos, in quibus via luminis non minima sed maxima est, immo alios quoque, in quibus neutra harum legum obtinet. Sistit praesens haec dissertatio nonnullas easque subitaneas in principium Lebnitianum, quatenus idem in Catoptricis adhibetur, animadversiones. Subitaneas inquam & sparsim collectas, non autem ejusmodi, quae ad plenariam & omnimodam principii commemorati discussionem pertinerent. Ea enim, qua premimur temporis angustia prohibuit, quo minus ex voto officio nostro vacare potuerimus. Fatemur etjam ingenue nos, occasione a praesenti argumento data, ad leves quasdam meditationes ab instituto aliquantum remotas, esse dilapsos; unde factum, ut nonnulla heic sint allata quae ad ipsam rem propositam si non nihil facere non tamen adeo necessaria videbuntur; multa etjam forte neglecta, quae adduxisse magis e re fuisset. Sed hujus defectus, eo facilius nos impetraturos esse veniam speramus, quod solummodo per specialia quaedam exempla falsitatem hypotheseos Lebnitianae ostendere voluerimus. Si de caetero Ben. Lectorum, nostrum qualemcunque laborem tuo favore dignatus, habebimus de quo nobis magnopere gratulabimur.
(a) Princip. Philosophiae P. III § 63 & 64. Dioptr. Cap. I § 8.
(b) Maupertuis essais de cosmologie pag. 41
(c) Vitellio Perspectiv. Prop. 20. Ptolemeus in Lib. I de speculis, Acta Erudit. Lips. Anni 1682. p. 185 nec non Anni 1701, p. 20
(d) Kraft Praelect. Phys. P. III. §§ 124, 170.
Generalis lex, secundum quam omnem luminis reflexionem fieri testantur experimenta, & ad quam exigendae sunt omnes hypotheses, quae ad reflexionem istam explicandam adhibentur, haec est, quod angulus reflexionis aequalis sit angulo incidentiae, uterque autem in plano communi, ad superficiem reflectentem perpendiculari. Primi hujus principi Catoptrici, quod communi legi reflexionis corporum elasticorum in obicem immotum impingentium, congruit, conventientiam cum hypothesi viae brevissimae ostensuri, tacite supponunt lumen in ipsam superficiem reflectentem impingere & quidem motu ferri rectilineo. Enim vero lumen, quod reflectitur, non impingere in partes solidas corporis reflectentis, argumentis magni ponderis evictum ivit Newton in Opticae Lib. II P. III. Prop. 8, qui & in Philos. Natur. Princip. Mathemat. P. I. prop. 96 hujusque Schol. maxima cum vero similitudine caussam reflexionis (atque ac refractionis, cfr. etjam Opticae Lib. I. P. I. prop. 6) derivat a vi repellente aut attrahente, quod corpora ad minutissimam aliquam distantiam in radios luminis agerent in lineis ad superficies suas perpendicularibus & quidem ad aequales distantias aequaliter; unde etjam fieret, ut radii, in accessu ad corpus atque recessu ab eodem, incurventur. Concedamus tamen posse istius incurvationis ut & negatae impactionis, in praesenti negotio, ubi de caussa reflexionis nulla instituitur quaestio, nullam haberi rationem; & ponamus potius radios luminis omnino in rectis lineis moveri atque ipsam superficiem reflectentem attingere. Quibus admissis, ex stabilita & ab omnibus physicis adprobata reflexionis lege, cujus supra mentionem facimus, dijudicari debet, num via luminis semper sit minima. Ostendemus autem viam luminis in superficies planas aut convexas incidentis, semper quidem minimam esse, at si de radiis a speculo cavo reflexis sermo sit, hoc adeo non universaliter valere; ut potius via ista tam maxima interdum, tum quoque nec maxima nec minima sit.
Ut itaque particulares nonnullos incidemus casus, in quibus Leibnitii principium aut locum habet aut non, primum attendere convenit ad viam luminis in superficies planas nec non convexas illapsi. Quem in finem sequentem adferemus propositionem. (Fig. 1). In superficie plana aut convexa determinare punctum C ita situm, ut rectarum AC CB ex datis duobus punctis A & B extra eandem positis, ad C ductarum summa sit minima, qua esse possit. Primum quidem evidens est, punctum illud C situm esse debere in plano, quod per A & B transiens, sit superficiei datae rectum, adeoque in communi illius plani atque hujus superficiei sectione.
Nam si plana fuerit superficies, & in ea aliud quoddam punctum K sumatur extra sectionem, quae jam erit linea recta fitque illa DE, ducta ad hanc normali KC & junctis KA KB CA CB, erit KC perpendicularis plano ACB ideoque & rectis CA CB (Euclid Elementa Lib. XI Defin. IV. III). In triangulis igitur rectangulis ACK BCK est AK > AC & BK > BC (Euclid Elementa Lib. I. 19 vel 47) hincque AK + BK > AC + CB. Ex quo patet, punctum illud non esse quaerendum extra sectionem DE, ergo in ipsa sectionem. Si superficies fuerit convexa, eadem ratione hoc demonstrari poterit. Sit igitur sectio illa aut linea recta DE, aut curva quaedam GCc convexitatem punctis A B obvertens. Sit in plano ABDE ad eam ac proinde ad ipsam quoque superficiem perpendicularis ducta CF, & AF BH perpendiculares ipsi CF. Sit puncto C aliud sectionis punctum c infinite vicinum quare Cc semper haberi poterit pro recta. Ductis porro Ac Bc centris A & B radiis AC & BC describantur arcus Cn, Cr, qui ad AC, Ac & CB normales erunt & ob infinitam suam parvitatem pro rectis habendi. Tunc quia angulus BHC = Crc uterque enim rectus, & rCc = HBC, nam HCB utriusque complementum est ad rectum; erit triangulum HBC ~rCc & proinde BH : BC :: Cr : Cc. Eodem modo ostenditur Δ AFC ~ Cnc. (Haec vero etjam, si placet, facile sequuntur ex Mac-Laur. Traité des Fluxions § 206.) Quando jam quantitas fluens AC + CB minima est, aequales erunt ipsarum AC & CB fluxiones videlicet cn = Cr, sicque AF : AC :: BH : BC, nec non (Euclid Lib. VI. 7) angulus ACF = BCH & proinde ACD = BCE. Oportet ergo punctum C ita simul situm esse, ut recte AC, BC efficiant cum superficie subjecta, vel & cum recta eidem in C perpendiculariter insistente, angulos aequales (e). Jam igitur, si fuerit ista superficies plana & proinde sectio linea recta DE, facile punctum istud C determinari potest, si secundum notissimam constructionem, ex alterutro datorum punctorum ut B in planum subjectum demittatur normalis (quae in sectionem cadit angulumque cum ea rectum efficiet Euclid XI. 38 & Defin. 3) BE producenda, ut sit EL = BE, & jungatur AL, haec enim occurret plano in puncto quaesito C. Ad hanc autem constructionem deducimur quoque investigato valore ipsus AF = DC = x vel EC = HB = a – x (si ponatur DE, quae jam datur, =a). Dantur nempe jam quoque AD, BE, quae dicantur b, c, respective. Et per demonstr. Δ AFC ~ BHC vel Δ ADC ~ BEC, erit b : x :: c : a – x indeque b + c : b :: a : x, ut & b + c : c :: a : a – x. Igitur x vel a – x per dictam constructionem facile inveniuntur. Si enim ducta intelligitur LN parallela DE, quae AD continuatae occurrat in N, est EL = DN adeoque AN (b + c) : AD (b) :: NL seu DE (a) : DC (x) quo ipso determinatur punctum C.
Quod sic quidem analytice invenimus, summam AC + CB minimam esse, si angulus ACF = BCH vel ACD = BCE, aut vicissim; id quoque ex primis Geometriae elementis facillime demonstratur; si superficies scilicet fuerit I:mo plane, eo modo, quo passim ex. gr. in Muschenbrockii Element. Phys. § 1036. Instit. Phys. § 1237. Kraftii Praelect. Phys. P. III. § 124 videre licet, si nempe ea his addantur, quae supra adducta fuere de puncto C extra sectionem non quaerendo. Si autem IIº convexa, sequenti ratione: In superficie convexa GCc sumto puncto quocunque I diverso a C, & junctis IA, IB, dico esse AC + CB < AI + IB. Quia superficies ponitur convexa, punctum I nunquam situm esse potest a parte AB plani DE eam in C tangentis, sed vel a parte opposita vel saltem in ipso hoc plano (f). Si hoc, erit AC + CB < AI + IB per Casum I:mum. Sin illud, AI + IB transibunt idem planum, in suo quaelibet puncto. Sit O punctum in quo AI plano isti occurrit, & jungatur BO; erit per Cas. I, AC + CB < AO + OB & (Euclid Elem. I. 20) AI + IB > AO + OB, quare a fortiori AI + IB > AC + CB.
(e) Quamvis dum curva GCc concavitatem punctis A B obvertit, analysis supra adhibita non minus valeat; eo tamen in casu punctum illud C sic definitum, non semper summam AC + CB minimam, sed etjam maximam determinat, uti in sequentibus ostendatur. Haec enim methodus per se aeque apta est ad determinandum maximum quid, ac minimum; utrum vero in casu speciali locum habeat, ex ipsius rei natura ac indole dijudicari debet.
(f) Posterius obtinet, si superficies fuerit cylindrica vel coniformis, dantur enim in illa puncta plano tangenti communia.
Constat ex § precedenti, si A B fuerint duo puncta ex quorum uno emissum lumen oporteat, reflexione in speculo plano vel convexa facta, ad alterum pervenire; ex supposita via luminis brevissima sequi aequalitatem angulorum incidentiae & reflexionis, aut vicissim ex hac aequalitate, viam illam maximam. Quemadmodum itaque hypothesis Leibnitii omnino vera est cum sermo fuerit de lumine, quod in unico speculo plano vel convexo reflexionem subit, sic eadem aequae locum habet, si repetatur reflexio in pluribus ejusmodi speculis. (Fig. 2). Sic extra specula aliquot plana, in diversis planis posita, dentur duo puncta. Oporteat radium lucis ex altero horum punctorum egressum, reflexione in singulis speculis facta, ad alterum pervenire. Quaerantur puncta incidentiae vel reflexionis ipsaque radii via. Sint primum duo specula FG, GH & puncta A, b, unum radians alterum illuminandum. Si radius ex A emissus in FG primum incidere debet, demissa ex A in FG perpendicularis AF producatur ab altera parte ut sit AF = Fa. Similiter ex puncto b demissa normalis ad GH (aut ejus planum, si opus sit continuandum, id, quod etjam de alio quovis casu simili dictum esto,) producatur donec Kβ fiat aequalis bk. Ducatur jam recta aβ; haec occurret speculis in punctis quaesitis C, D, & junctis AC, bD habetur via radii ACDb = aβ (g). Quum scilicet the angle AFC = aFC & aF = AF, erit (Eucl. I. 4. 15.) angulus ACF = (aCF =) DCG; & similiter ang. bDK = CDG, id est anguli reflexionis utrobique aequales sunt angulis incidentiae. Porro quia aC = AC, bD = βD, erit AC + CD + Db = (aC + CD + Dβ =) aβ. Sint jam tria specula FG GH HI, puncta autem data A B. Determinato ut antea puncto a demissoque ex B ad speculum HI, in quo ultima fiat reflexio, catheto reflexionis ad b producendo ita ut IB = Ib, iterum ex b ad GH demissum perpendiculum producetur ad β ut sit Kβ = Kb. Jam ducta aβ dabit puncta C & D desiderata, & juncta porro Db determinabit punctum quaesitum E in speculo HI. Tumque ductis AC, BE, erit radii via ACDEB = aβ, quod pari ratione ac supra ostenditur.
Quodsi autem specula adhuc, fuerint plura, similis sed repetita adhibeatur operatio, quae quemadmodum istituenda sit, ex dictis facile intelligitur. Eandem quoque solutionem locum habituram esse apparet, si puncta A, B (vel b) coincidant, hoc est, si lumen ex A egressum ad idem, unde exierat, punctum redire debeat. In omnibus autem his casibus, est via luminis semper minima. Nam si ponatur radium luminis in alia quaecunque speculorum puncta ut c d impingere, ex A e.g. ad b perventurus, via ipsius Acdb > foret ACDb. Est enim (Eucl. Elem. I. 20.) ac + cd + dβ > aβ, adeoque propter Ac = ac & bd = βd, Ac + cd + db > AC + CD + Db. Idem valet, si consideremus viam luminis, ut ACDEB, in pluribus speculis reflexi.
(g) Supponitur heic, lineam aβ occurrere speculis, FG, GH, hoc enim nisi obtineret, impossibile esset, lumen reflexione ejusmodi, qualis supposita fuit, ex A ad b pervenire.
Examinata reflexione in speculis planis & convexis ad concava accedimus. Et primum quidem ut de via luminis a speculis cavis sphaericis reflexi pronunciari queat, una alterave propositio per modum lemmatum praemittenda videtur. Prima autem haec esto (Fig. 3): In diametro circuli, vel eadem producta, duobus punctis ut cunque sumtis, se aequae a centro distantibus, quadrata rectarum ab his punctis, ad quodcunque punctum peripherae ductarum, ubique efficient summam constantem (h) aequalem scilicet quadratis semidiametri & distantiae alterutrius istorum punctorum a centro bis sumtis, aut vero summae quadratorum ex segmentis diametri, in quae ab alterutro istorum punctorum dividitur. Sit EIF circulus in cujus diametro EF puncta A B sumta sint aeque a centro C distantia, ex quibus ad quodcunque peripheriae punctum G ductae sint rectae GA, GB. Ducatur GC & demittetur ex G ad diametrum perpendicularis GD. Igitur quia AGq = ACq + CGq + 2·AC · CD & GBq = GCq + CBq – 2·CB · CD (Euclid Elem. II 12. 13.); erit ob AC = CB, summa AGq + GBq = 2·CGq + 2·ACq adeoque constans, dato nimirum circulo atque punctis A, B. Ulterius quia CG = CF, erit CGq = CFq unde AGq + GBq = 2·CFq + 2·ACq; sed (per Euclid II. 10. vel 9.) est AFq + BFq seu AFq + AEq = 2·ACq + 2·CFq; ergo AGq + GBq = AFq + AEq (i).
(h) Curvam, cujus haec sit proprietas, circulum esse, mutata paululum serie demonstrationis in hac §pho adhibita, facile probatur. Scilicet propositum esto: datis punctis A & B, invenire curvam EGF, a cujus puncto quocunque G ad A & B ductarum rectarum quadrata efficiant summam constantem. Jungatur AB, in quam, productam si opus est, demittatur perpendicularis GD; bisecta sit AB in C & jungatur CG. His praemissis, demonstrabitur ut in hac §:pho, esse GAq + GBq = 2·CGq + 2·CAq. Adeoque quum ex hypothesi GAq + GBq sit constans, constans etjam erit 2·CGq + 2·CAq vel ejus dimidium CGq + CAq. Quare quum detur CA ideoque CAq, constans quoque erit reliquum CGq, consequenter recta CG a dato puncto C ad quolibet curvae punctum ducta, constantis erit magnitudinis. Unde patet curvam quaesitam esse circulum cujus centrum est C radius autem CG. Si calculo uti placuerit, dicantur AB a, CD x, GD y; erit AD = ½ a + x, BD = ½ a – x vel x – ½ a; & sint AGq + BGq = dato quadrato bb. Igitur AGq = (ADq + GDq =) ¼ aa – ax + xx +yy, consequenter AGq + BGq seu bb = ½ aa + 2 xx + 2 yy unde yy + xx = ½ bb – ¼ aa, quae est aequatio ad circulum. Curva igitur quaesita est circulus centro C, radio = √(½ bb – ¼ aa) hunc in modum describendus. Ex A transferatur, in rectam CA productam, AK = ½ b cui equalis & perpendicularis fiat KH. Tum centro A intervallo AH describatur arcus perpendicularem ex C erectam secans in I. Si jam centro C radio CI describatur circulus, satisfaciet hic problemati.
(i) Sistit praesens propositio casum maxime simplicem & specialem, contentum sub generali & admodum memorabili circuli proprietate, in Marchionis Hospitalii Analyse des infiniment petits § 33. demonstrata.
Altera propositio praeliminaris haec esto (Fig. 4): Inter triangula innumera ABC super eadem basi AB constituta & datum habentia angulum ACB basi oppositum, quaeritur illud, quod summam crurum AC + BC maximam habere. Producta sit BC versus D usque dum fiat CD = CA, consequenter AC + CB = BD, & jungatur AD. Erit itaque (Eucl. I. 5. & 32.) angulus ACB duplus anguli D. Quare propter datum ang. ACB, datus etjam est D. Omnia itaque puncta D per (Eucl. III. 21. convers.) collocantur in circumferentia circuli ADEB, cujus segmentum super recta AB constitutum capit angulum datum, aequalem scilicet dimidio dati ACB.
Omnium vero cordarum BD in hoc circulo maxima est (Eucl. III. 15.) diameter. Quando autem BD est ipsa diameter non potest non punctum ejus C esse (Eucl. III. 7.) circuli centrum, proter aequales (ex construct.) CA, CD. Erit itaque in hoc casu, ubi nimirum BD seu AC + CB maxima est, CA = CB, h.e. triangulum Isosceles prae omnibus, quorum eadem est basis idemque angulus basi oppositus, summam crurum maximum habet.
Hoc ipsum vero etiam per methodum de maximis et minimis (k) calculo fluxionum super structam, facile invenietur. Sit basis AB constans = b, AC = x – CB = y. Ducta concipiatur in crus unum BC, productum si opus est, ex vertice anguli ei oppositi perpendiculum AF. Ob ang. C constantem, constans est ratio inter AC & CF (Euclid VI. 4.) quae ponatur = l:n, quare CF = nx. Jam vero (Euclid II. 12. vel 13.) ABq = ACq + CBq ± 2· CB · CF, i.e. xx + yy ± 2 nxy = bb. Hinc sumendo fluxiones, & dividendo omnes terminos per 2, erit x dx + y dy ± ny dx ± nx dy = 0. Quando jam summa AC + CB ponitur maxima, est ejus fluxio dx + dy = 0. Quare pro dy substituendo – dx & dividendo per dx, sit x ± nx = y ± ny, & dividendo per 1 ± n, x=y. Est ergo Δ:m quesitum Isosceles, facile hoc modo construendum. Super data basi AB describatur segmentum circuli, quod capiat angulum dato aequalem, & arcu istius segmenti, qui est locus verticum ΔΔ:orum illorum numero infinitorum, bisecto, habetur vertex quaesiti (Euclid III. 33. 21. 30.). Hinc autem vicissim colligere licet, ex omnibus ΔΔ:lis eandem crurum summam, aequalem angulos comprendentiam habentibus, aequicrurum basi insistere minimae, (l) quod vero etjam novo calculo invenitur. Sit summa illa a, crus AC = x, BC = a – x, CF ut antea = nx, & ACq + CBq ± 2· CB · CF = ABq hoc est aa – 2ax + 2xx ± 2nax – (± 2nxx) = ABq. Posita jam AB omnium minima, erit etiam ipsius quadratum minimum; ideoque hujus fluxio = (–2a + 4x ± 2na – (± 4nx))· dx =0 unde dividendo per 2 dx & transferendo terminos, sit 2x – (± 2nx) = a –(± 2na) & x = (±na–a)/ (±2n–2) =½ a = a – x.
(k) In praesenti quidem negotio facile intelligitur hac methodo determinari casum, in quo summa crurum maxima sit, nec omnino dari minimam, sed decrescere illam posse usque dum excessus ipsius supra basin minor fiat quavis assignabili quantitate adeoque ipsum Δ:lum evanescat (Eucl. I. 20. 22.),
(l) Vicissim evidens est, data summa crurum cum angulo iis comprehendo, non dari casum, in quo basis maxima sit, sed posse hanc tantam esse, ut quantumvis prope ad ipsam crurum summam accedat.
Consideraturi jam casus nonnullos circa reflexionem luminis, in speculis cavis sphaericis factam, primum ostendimus (Fig. 3): Sumtis in diametro EF speculi concavi sphaerici, duobus punctis A & B a centro aeque distantibus, radios ab uno eorum emissos & ad alterum reflexos viam impendere longissimam, exceptis tamen illis AF + FB, vel AE + EB, qui reflexi in semet ipsos redeunt (*), hi enim brevissima feruntur via. In circulo sphaerae maximo quocunque per EF transeunte, (quippe ad speculum normali) diameter supra EF perpendiculater erectus dabit in circumferentia circuli puncta I incidentiae luminis (id quod ex aequalitate angulorum CIA CIB per Eucl. I. 4. manifeste sequitur); adeo ut AI + IB fit via radii incidentis ac reflexi, quae quidem erit longissima. Nam si descripta intelligatur Ellipsis focis A & B, per I transiens, cadat illa tota extra circulum, quippe quae habet centrum C & semiaxem minorem CI. Ductis jam ex A B ad punctum quodcunque circuli a I diversum ut G, rectis AG, GB, quarum una ut AG concipiatur producta donec occurrat ellipsi in O, & juncta OB, erit (per propriet. ellipseos) AI + IB (seu AO + OB) > AG + GB. (Eucl. I. 21.) Idem sic quoque evincitur. Concipiatur triangula rectangula, quorum crura sint AI IB & AG GB; habebunt illa hypothenusas aequales (Eucl. I. 47.) quoniam (§4) AIq + IBq = AGq + GBq. Ergo quum sit AI = IB, erit (vi. §5) AI + IB > AG + GB. Et quidem summa AG + GB eo evadit minor, quo longius ab aequalitate recedunt AG & GB i.e. (Eucl. III. 7.) quo proprius est punctum G ipsi E vel F, in quae puncta EF cum inciderint, erit AF + FB scilicet via radii, qui in seipsum reflectitur, minima omnium AG + GB. Potest quoque haec propositio sequenti ratione demonstrari. Sumto alio puncto G in circulo, producatur AG ad L donec AL fiat = 2 AI vel AI + IB, & bisecetur AL in N ut sit AN = AI. Quare propter (Eucl. III. 7.) AG > AI erit AG > AN; & quidem quo proprior fuerit AG diametro EF, eo major (Euclid loc. citat.) fit AG adeoque etjam NG. Et quia AL in N secta est in partes aequales, in G autem in inaequales, erunt (Eucl. II. 9.) AGq + GLq = 2·ANq + 2·NGq; vel ob 2·ANq = (2·AIq = 2·AIq + 2·IBq = §4) AGq + GBq, erit AGq + GLq = AGq + GBq + 2·NGq, consequenter GLq = GBq + 2·NGq; adeo ut semper sit GL > GB ideoque addita communi AG, AL seu AI + IB > AG + GB. Ulterius 2·NGq = (GLq – GLq = Eucl. II. 5. cor.) GL + GB · GL – GB. Prout jam punctum G propius accedit ad F extremitatem diametri EF sibi propiorem; ob crescentem tunc (per demonstr.) NG, constantes vero AL & AN, ideoque decrescentes GL & (Eucl. III. 7.) GB ac proinde GL + GB crescit 2·NGq & ipsi aequale (per demonstr.) rectangulum sub GL + GB & GL – GB contentum; ergo (ob decrescentem GL + GB per demonst.) crescit GL – GB, adeoque crescit etiam ipsius (AG + GL seu) AI + IB excessus supra AG + GB vel AG + GB minor fit continue donec fiat AF + FB, quae proinde minima erit. Adparet vero omnes radios ex A emissos, in peripheriam circuli sphaerae maximi, diametrum EF perpendiculater secantis, incidentes ad B reflecti & per viam longissimam moveri; ista enim peripheria locus est omnium punctorum I.
(*) Hoc autem locum habere nequit, nisi puncta ista A B cadant intra sphaeram.
Quod in § praecedenti demonstravimus, viam luminis maximam dari, cum in ipsa diametro speculi cavi sphaerici sumuntur puncta a centro aequae distantia radians & illuminandum; idem in variis etjam casibus aliis valere censendum est (*). Ita (Fig. 5) Sumtis duobus punctis R, S, in superficie speculi cavi sphaerici, radius ex uno horum punctorum esmissus & per reflexionem a speculo factam ad alterum perveniens, viam impendit longissimam, si reflexio, fiat ad easdem partes plani per RS, quod plano sphaere circuli maximi per R S transeunti rectum sit, ad quos cadit centrum sphaerae, sin ad oppositas, via luminis nec maxima nec omnino maxima est. Jungatur RS super qua in K bifariam secta erigatur normalis IT in plano circuli sphaerae maximi per RS transeuntis, quare erit IT ad sphaerae superficiem perpendicularis; occurrat autem peripheriae hujus circuli in binis punctis I & T in quibus punctis reflexionem fieri debere ex leges reflexionis manifestum est, cum (Euclid I. 4.) in ΔΔ RIK, SIK sit ang. RIK = SIK, & simili ratione ang. RTK = STK, recta autem IT sit diameter (Eucl. III. 1. Cor.) ac proinde normalis ad circulum (Eucl. III. 18.). Sumto jam primum quidem in circumferentia circuli ad eandem partem rectae RS ad quam cadit centrum circuli sive sphaerae & ad quam scilicet situm sit I, alio quocunque puncto G & ductis RG, SG, erit RI + IS > RG + GS. Quum enim ang. RIS = RGS (Euclid III. 21.) RI autem = sit IS sequitur inde (§5) in Δ:o aequicruro RIS summam crurum RI + IS esse > RG + GS. Deinde assumto alio quolibet superficiei sphaericae puncto O in circumferentia circuli RIS non posito, & ductis RO SO, dico adhuc etjam fore RI + IS > RO + OS. Secetur enim sphaera plano per ROS, eritque sectio circulus sphaerae minor, qui circa RS revolvatur donec planum ejus in planum circuli maximi RIST cadat & punctum quidem O ad partes I plani praedicti per RS; quo facto facile intelligitur, circuli illius minoris, quippe maximum in RS secantis, segmentum ROS cadere debere intra circumferentiam RIGS, reliquum vero segmentum SNR extra reliquam circumferentiam STR. Sumatur itaque in RIGS aliquod punctum G tale; ut junctis RG GS, O cadat intra Δ RGS; unde erit (Euclid I. 20.) RO + OS < RG + GS; at (per demonstr.) RI + IS > RG + GS, ergo multo magis RI + IS > RO + OS. Ostendum itaque est, esse RI + IS summam maximam rectarum a punctis RS ad quodlibet in superficie sphaerae punctum ducendarum. Deinde quod ad alterum casum attinet; sumto primum in circumferentia RTS alio ut lubet puncto H praeter T, & ductis RH HS; eodem, quo supra dictum modo patet (§5) esse RT + TS > RH + HS adeoque non minimam, sed respective maximam. Non tamen absolute maxima est. Si enim circulus quicunque sphaerae minor per RS transiens revolvi concipiatur circa rectam RS immotam, usque dum in planum ipsius RIST cadat illius circuli minoris. Ut supra jam monitum, utrum libet super recta RS constitutum segmentum cadat extra circumferentiam RTS. Cadat ut RNS; quare sumi poterunt in RNS puncta N ejusmodi, ut ductis rectis RN, NS, T cadat intra Δ RNS, ideoque sit RT + TS < RN + NS. Ergo RT + TS nec minima nec prorsus maxima est. (Fig. 5) Si autem PQ, ex quorum uno P egressum lumen, reflexione in speculi puncto I facta, ab alterum Q pervenerit, eum obtineant situm vel intra vel extra superficiem sphaerae, cujus portionem constituit speculum, ut utrumque istorum punctorum PQ, respectu ipsius I, cadat ad oppositam partem diametri EF, quae in circulo sphaerae maximo per PIQ transeunte, parallela ducitur rectae eundem in puncto incidentiae I tangenti; rursus longissima erit luminis via PI + IQ. Nam ducta per I semidiameter IC perpendicularis erit (tangenti per I adeoque rectae quae huic parallela est videlicet) ipsi EF; quare, si occurrant RI IS diametra EF (productae si opus sit) in A B punctis, in ΔΔ ICA ICB, propter angulos ad C rectos & angulos ad I (per leg. reflex.) aequales, erit (Eucl. I. 26.) AC = CB; hinc sumto alio quocunque, praeter I, peripheriae EIF puncto G & junctis AG BG, erit (§6) AI + IB > AG + GB, unde additis utrinque AP + BQ, fiet PI + IQ > AP + AG + GB + BQ. Est vero (Euclid I. 20.) AP + AG > PG & GB + BQ > GQ ergo a fortiori erit PI + IQ > PG + GQ. Atque probari etiam posset PI + IQ majorem esse rectis simul sumtis, quae e punctis P Q ad punctum quodcunque speculi extra circumferentiam EIF assumtum ducuntur; enimvero ut nimia vitetur prolixitas, hac demonstratione supersedemus, & sufficiat jam ostendum esse, quod via luminis IP + PQ brevissima haud sit.
Eandem hanc propositionem Milliet Dechales in Mundo Mathemat. Tom. III. Catopt. Lib. I. p. 570 facillima demonstratione probavit.
Missis igitur speculis cavis sphaericis, casus simplicissimos reflexionis factae in speculis cavis revolutione Sectionicis cujuscunque Conicae circa axem formatis, considerabimus. Et primum quidem quod ad specula parabolica aut hyperbolica attinet, radius luminis e foco F speculi parabolici aut hyperbolici egressus, & ad punctum quodcunque I reflexus aut vicissim, via fertur brevissima. Secetur scilicet conoides hoc parabolicum aut hyperbolicum secundum axem AD, plano per I transeunte, quippe in quo erit radius tam incidens, quam reflexus per §1, eritque sectio parabola aut hyperbola scil. Apolloniana, cujus focus est F. Dico primum, sumto extra hanc sectionem puncto speculi quocunque, dicatur v & per mentem ductis Fv, vI, non posse summam Fv + vI esse minimam. Si scilicet Fv concipiatur revolvi circa axem curvae, describet punctum v peripheram circuli, per cujus centrum transit AD & plano ejusdem circuli perpendiculariter insistit. Postquam (quod fieri debere genesis conoides loquitur) v sectionem ABG attigit, in qua Fv situm FV obtinuit, jungatur IV, quae utpote in eodem plano cum axe AD existens, cum illa aut convergit, aut ipsi parallela est. Si, hoc ponatur, erit IV ad planum circuli perpendicularis (Eucl. XI. 8.) adeoque juncta per mentem Vv, IVv angulus rectus, quare IV < Iv. Sin vero illud, producatur VI donec axi occurrat in puncto P; & si juncta intelligatur Pv, erit PV = Pv (vid fis Wolf Elem. Geomet. §487). Ergo quia in Δ:lo PIv (Eucl. I. 20.) Pv + vI > PI (Fig. 6), vel (Fig. 7 PI + Iv > PV) erit PI – PV (vel PV – PI) id est IV rursus < Iv. Hinc FV + VI < Fv + vI. Unde adparet summam Fv + vI non posse esse minimam. His praemissis, in parabola ducatur per I diameter IB parabolae occurrens in puncto B, quod erit punctum incidentiae, juncta autem FB, FBI ipsa luminis via, ceu ex proprietate parabolae & lege reflexionis satis constat. Dico illam esse brevissimam seu FB + BI < FV + VI, sumto in curva alio quocunque puncto V & ductis FV, VI. Nam ducatur diameter VE, cui in E occurrat recta ID, quae per I ad axem AD perpendicularis ductitur. Producatur IB, EV, ut occurrant directrici CG; propter parallelas IG EC nec non GC IE, erit IG = EC (Eucl. I. 34.). Adeoque quum ex natura parabolae sit FB = BG, & FV = VC erunt FB + BI = (IC = EC =) FV + VE, adeo ut omnes radii e foco speculi parabolici egressi & ad idem planum axi perpendiculare reflexi, aequalem absolvant viam. At in triangulo rectangulo VEI, latus VI > VE (Eucl. I. 19. vel 47.) hincque FV + VI > (FV + VE =) FB + BI. Quod vero ad hyperbolam attinet (Fig. 7); e foco ejus externo, qui sit f ducatur ad punctum I recta fI, occurrens hyperbolae in B, erit per proprietatem hyperbolae & legem reflexionis, B punctum incidentiae vel FB + BI via radii incidentis & reflexi. Sit V aliud quodcunque punctum curvae, ductis FV VI, erit FB + BI < FV + VI. Ducta enim per f V recta, quae in C occurrat circulo DICG centro f intervallo fI descripto, est fB + BI = fV + VC & per naturam hyperbolae fB – FB = (axi traverso =) fV – FV, qua aequatione subducta a priori erit FB + BI = FV + VC. Sed (Euclid III. 7.) VC < VI, ergo FV + VC vel FB + BI < FV + VI. Ex allatis etjam sequitur, radios omnes e foco speculi hyperbolici cavi egressos & a speculo reflexos, ad superficiem sphaericam, cujus centrum est focus hyperbolae oppositus, perveniendo aequalem conficere viam = AD + AF, si FD sit semidiameter sphaerae.
Verum (Fig. 8) Si e foco alterutro F speculi concavi elliptici emittantur radii ad punctum A reflectendi, brevissima huc via perveniet is, qui ab eandem axeos majoris parte movetur ad quam situm est A, longissima vero, qui ab altera. Necesse est, ut puncta speculi, in quibus reflexio fit radiorum, sint in plano per puncta F A transeunte & ad speculum ellipticum perpendiculari §1 adeoque in sectione superficiei ellipsoidis, plano secundum axem BD & per A sectae, quae sectio BCD itaque erit ellipsis scil. Apolloniana. Ducta per punctum A & alterum focum, qui sit f, recta bis occurrent ellipsi, determinabit bina illa puncta C, c, in quibus (posito A esse intra ellipsoidem) supposita ista reflexio peragitur. Ductis proinde FC, Fc, habentur viae radiorum quaesitae FC + CA & Fc + cA, id quod ex natura ellipseos & lege reflexionis fluit. Dico jam viam FC + CA minimam esse, sed Fc + cA maximam. Scilicet sumto alio quocunque puncto V sive in curva sectionis sive extra eandem & ductis VA VF Vf, erit FC + CA < FV + VA. Etenim quia per proprietatem ellipseos FV + Vf = FC + Cf atqui VA + Af > Vf (Eucl. I. 20.), erunt FV + VA + Af > FC + Cf adeoque demta utrinque communi Af, erit FC + CA < FV + VA, adeoque minima, quae esse possit. Ex adverso autem dico, esse Fc + cA maximam. Nam FV + Vf = Fc + cF, quare addito utrinque fA, FV + Vf + fA = Fc + cA. Sed (Eucl. I. 20.) Vf + fA > VA ideoque Fc + cA > FV + VA. Consequitur ergo Fc + cA esse viam maximam. Patet vero simul differentiam viae longissimae & brevissimae aequalem esse 2Af, quoniam FC + CA + Af = Fc + cA – Af (videlicet FC + Cf = Fc + cf per propriet. ellips.) indeque FC + CA + 2Af = Fc + cA. Catereum evidens est, si speculum cavum, ut VCD fuerit portio quaedam superficiei ellipsoidis, & extra integrae hujus ambitum positum sit punctum quodcunque a, lumen e foco F ad hoc punctum a reflexum aut vicissim non nisi unica eaque longissima via FC + CA pervenire. Ductis enim FV, VA, Vf, est FC + Cf = FV + Vf adeoque addita fa, erit FC + Ca (= FV + Vf + fa) > FV + Va (Eucl. I. 20.).
Quodsi in axe speculi cavi Elliptici, revolutione scilicet Ellipseos circa alterutrum axem suum formandi (Fig. 8), sumantur duo puncta H I utrinque a centro aeque distantia, & quidem, si axis speculi fuerit axis major Ellipseos, ultra focos sita; lumen, quod ex altero illorum punctorum emissum, reflexione ad alterum pervenit, maximam emetiri viam, praeter quam in eo casu, quo lumen in se ipsum, si fieri possit, reflectitur, pari ratione demonstratur, qua de speculo cavo sphaerico idem supra ostendum est, §6 circa initium. Neque minus quae in §7 de speculo sphaerico dicta sunt, niutatis nonnihil aut novis additis determinationibus, ad speculum ellipticum adplicari possunt.
Descripta (Fig. 8) concipiatur focis F & f atque semiaxe minore EG ellipsis BGD. Esto GLM Sectio quaecunque Conica axem habens (transversum quidem, si de hyperbola sermo fuerit,) GE positione, verticem G, sitque latus rectum ad hunc axem pertinens minus latere recto axis minoris Ellipseos BGD. Quoniam igitur omnis sectio conica in vertice axis eandem habet curvaturam cum circulo, cujus diameter aequalis est parametro ejusdem axis; sequitur inde Curvae GLM, quippe in G minorem quam ellipsis BGD curvaturam habentis, saltem aliquam circa G portionem ut GL intra Ellipsin cadere. Quamobrem FG + Gf superat summam rectarum e punctis F & f ad idem quolibet punctum arcus GL ducendarum. Neque ideo tamen FG + Gf excedit summam rectarum, quae ex F & f ad punctum curvae GLM utcunque assumptum duci possunt; nisi haec tota intra ellipsin BGD ceciderit. Et quidem si GLM fuerit Parabola vel Hyperbola, summam ejusmodi rectarum continue majorem sumere licet, cum hae curvae ad distantiam ab F & f quavis data majorem extendi queant. Si jam fuerit GLM speculum concavum, vel potius sectio facta plano per axem speculi, adeoque ad hoc ipsum perpendiculari; lumen quod ex alterutro punctorum F f exiens, in speculi punctum G incidit, ad alterum reflecti patet. Atque ex dictis constat, hanc luminis viam FG + Gf neque brevissimam nedum longissimam esse. Quin imo, cum possibiles sint aliae curvae infinitae, axem habentes GE ejusque verticem G, in quo tangens sit axi perpendicularis, & quae concavae sint versus axem atque ab eodem longius continue recedant cruribus divergentibus, quarum vero portio aliqua intra ellipsin BGD cadat; evidens est vel ex isthoc capite casus dari innumeros, in quibus luminis ex F ad f, aut vicissim reflexionem in speculis cavis subeundo, pervenientis via nec minima est nec maxima.
Vidimus itaque principium viae brevissimae in Catoptrica, etjamsi veritati saepe consentaneum (§§ 2. 3. 8.) multis tamen exceptionibus esse obnoxium, videlicet lumen saepe longissimam impendere viam (§§ 6. 7. 9. 10. 11.) & quidem interdum non nisi longissimam (§§ 6. 9. 10.) nonnumquam vero vel brevissimam vel longissimam (§§ citt.); dari denique casus sequentes, in quibus via illa nec minima sit nec absoluta maxima (§§ 7. 11.), idque sive sermo fuerit de distantiis, quibus (§11) puncta speculi in ipsa ejus sectione (facta plano per puncta radiens & illustrandum ad speculum perpendiculari), sive quibus singula universim speculi puncta (§7.) a binis hisce punctis datis distant. Ultimum hoc unico adhuc exemplorum genere facillimo, plures casus singulares complectente, comprobare lubet. Sit speculum cavum cylindriforme, motu parallelo vel circuli vel ellipseos, vel alius curvae ejus indolius ac in §11 diximus, secundum lineam rectam plano ipsius perpendicularem formatum, adeo ut quaelibet sectio communis speculi atque plani, quod curvae generanti parallelum ideoque speculo perpendiculare est, huic ipsi curvae prorsus aequalis & similis est. Sumtis jam in plano cujuscunque hujusmodi sectionis duobus punctis radiante & illustrando, quorum situs respective determinetur eo modo, quo ipsorum A & B in §6, H & I §10, P & Q vel R & S §7, F & a §9, F & f §11; ex §§ citatis constat dari viam luminis, quae, dum ratio habetur punctorum speculi in ipsa illa sectione positorum, maxima (saltem §11 non minima) erit. Eandem tamen absolute maximam haud esse, vel inde mox adparet, quia sumto in recta illa linea, quam durante motu praedicto describit in superficie speculi ipsum punctum reflectens, puncto aliquo X, hujus a punctis radiante & illuminando distantiae majores continue fiant, prout remotius fuerit X a puncto reflectente. Scilicet hypothenusa maximum est laterum Δ:li rectanguli; atque manete uno crurum, altero autem crescente, cresit etjam hypothenusa.
Pertinet allatum jam exemplorum genus ad specula cava, revolutione curvae alicujus circa axem suum non formenda. Quod vero neque in speculis, quae rotatione ejusmodi orta concipiuntur, via luminis semper vel absolute maxima sit vel minima, exemplum loquitur in §7 memoratum, ubi RT + TS non omnino maximam nedum minimam esse ostendum fuit.
At ne quid dissimulemus, etjam mentio nobis injicienda videtur alicujus refugii, quo defensores hypotheseos Leibnitianae omnes difficultates, quibus suum principium premi forsitan intellexere ipsi, eludere conantur. Scilicet ea est, qua Dechales (*) a semet ipso allatae objectioni contra hypothesin viae minimae obviam ire vult, quod nempe speculum curvum non debeat considerari ut unicum aliquod speculum, sed ut ex infinitis planis speculis compositum. In qua explicatione ipsum Leibnitium quoque praesidium quaerere velle, innuere obscurius licet, videntur verba, quibus adfirmat suam hypothesin ad superficies concavas aut convexas adplicari, considerando earum planas tangentes (**). Verum quicquid sit, videtur haec responsio revera non adeo magni esse momenti. Praeterquam enim quod adcurate loquendo superficies curva vix ac ne vix quidem possit considerari, ut ex infinitis planis oriunda, cum potius ipsa genesis curvarum loquatur, quamlibet partem earum, utcunque exiguam, reapse esse curvam, etsi obsummam parvitatem quoque pro recta haberi queat; non satis equidem intelligo quomodo, etiam hac admissa explicatione, Leibnitii hypothesis in salvo ponatur. Etjamsi enim verum sit, viam luminis, a speculo cavo reflexi, minorem esse summa rectarum, quae ex puncto radiante & illustrato ad diversum a reflectente punctum plani, speculum in puncto reflectente tangentis, duci possunt; hoc tamen non aequae valet, si comparatio instituatur inter viam luminis & summam ductarum ex his punctis ad alium quoddam curvae seu caeterorum planorum tangentium punctum. Adeo ut supposita explicatione ista, frustranea omnino esset adplicatio principii commemorati ad specula concava. Quatenus enim haec ut plana spectantur, locum quidem haberet, non item si tanquam curva considerentur. Quodsi autem visum fuerit idem principium ad specula cava adplicandum, hunc in modum intelligere atque enunciare: Reflexionem luminis fieri in illo speculi puncto, in quo si planum tangat superficiem speculi, ejusdem puncti distantiae a punctis radiante & illustrando simul sumtae minores sint horum distantiis ab alio quovis ejusdem plani tangentis puncto; veritatem ejus facile concedimus. Quemadmodum vero hunc sensum ex citatis Leibnitii verbis non tamen facile exsculpseris, ita nec ratio adparet cur specula convexa in eundem ac concava censum ipsi veniant, cum de illis valeat hypothesis viae brevissimae absolute posita (§2) de his non item. Caeterum si cui videbuntur ea, quae in his pagellis disputata sunt, universalitatem principii Leibnitiani infringere, eo ipso simul fatebitur vacillare etjam principium actionis minimae ab Illust. de Maupertuis propositum. Quum enim quantitas actionis aestimetur facto ex massa corporis moti in celeritatem & spatium percursum, massa autem & celeritas luminis in negotio reflexionis maneant invariatae, adparet
principium actionis minimae hoc in casu coincidere cum principio viae minimae,
adeoque quae adversus hoc recte moventur dubia,
etjam contra illud valere.
* Mund. Mathemat. Tom. III. Catopt. Lib. I. p. 570
** Acta Eridutor. Lipsiens. A. 1682. p. 185.