The Rays In All Ways That Serve The View

How can these artificial organs be made to make them as perfect as possible?

First, I explain the shapes that the surfaces of transparent bodies must have in order to bend and divert the rays of light in all the ways that may serve my purpose.

I will use ellipses or hyperbolas, circles, and straight lines.

The ellipse or oval is a curved line which mathematicians create by cutting across a cone or a cylinder. I have also seen this used by gardeners in the compartments of their beds, where they describe it in a way that is really very crude and not very exact, but which, it seems to me, makes its nature better understood than the section of the cylinder or of the cone.

They plant 2 stakes in the ground, one at point H and at point I. Then they tie these together as the two ends of a cord. They pass it around them in the way you see here BHI.

Then, putting the tip of their finger in this rope, they lead it all around these two stakes, always pulling it towards them with equal force, in order to keep it also taut, and thus describe on the earth the curved line DBK, which is an ellipse.

If, without changing the length of this cord BHI, they only plant their posts H and I a little closer to each other, they will again describe an ellipse, but which will be of a different kind than the preceding one.

If they plant them a little closer still, they will describe yet another; and finally, if they join them together completely, it will be a circle that they will describe; whereas, if they decrease the length of the cord in the same proportion as the distance of these pegs, they will describe many ellipses which will be different in size, but which will all be of the same kind.

And so you see that there can be an infinity of species, all different, so that they differ no less from each other than the last forms the circle, and that of each species there is there can be any sizes.

If, from a point, like B, taken at discretion in one of these ellipses, we draw two straight lines towards the two points H and I, where the two stakes must be planted to describe it, these two lines BH and BI, joined together, will be equal to its greatest diameter DK, as is easily proved by construction; for the portion of the cord which extends from I towards B, and from there folds up to H, is the same which extends from I towards K or towards D, and from there also folds up to H, in so that DH is equal to IK; and HD plus DI, which are equal to HB plus BI, are equal to mowing DK; and finally, the ellipses which are described by always putting the same proportion between their greatest diameter DK and the distance from the points H and I, are all of the same species.

Because of a certain property of these points H and I, which you will hear below, we will call them the hot points, one interior and the other exterior: namely, if we relate them to half of the ellipse which is towards D, I will be the exterior; and if we relate them to the other half which is towards R, it will be the interior; and when we speak indiscriminately of the burning point, we will always hear of the interior.

Then, besides that, you need to know that if, through this point B[36], we draw the two straight lines LBG and CBE, which intersect each other at right angles, and one of which LG divides the angle HBI into two equal parts, the other CE will touch this ellipse at this point B without cutting it: of which I do not put the demonstration, because the geometers know it well enough, and the others would only do so bored to hear it.

But what I particularly intend to explain to you here is that, if we draw again from this point B, outside the ellipse, the straight line BA, parallel to the greatest diameter DK, and that, the having taken equal to BI of the points A and I, one draws on LG the two perpendiculars AL and IG, these two last AL and IG will have between them the same proportion as the two DK and HI.

So that if the line AB is a ray of light, and that this ellipse DBK is on the surface of an entirely solid transparent body, through which, according to what has been said above, the rays pass more easily than through the air in the same proportion as the line DK is greater than HI: this ray AB will be so diverted to the point B, by the surface of this transparent body, that it will go from there towards I.

And, inasmuch as this point B is taken at discretion in the ellipse, all that is said here of the radius AB must be understood generally of all the rays, parallel to the axle DK, which fall on some point of this ellipse, namely that they will all be there so diverted that they will go from there to point I.

This is demonstrated in this way.

1. If we draw from point B the line BF perpendicular to KD, and if from point N, where LG and KD intersect, we also draw the line NM perpendicular to IB, we will find that AL is at IG, as BF is at NM.

For, on the one hand, the triangles BFN and BLA are similar, because they are both right-angled, and because NF and BA being parallel, the angles FNB and ABL are equal; and, on the other hand, the triangles NBM and IBG are also similar, because they are right-angled, and the angle towards B is common to both.

Besides this, the 2 triangles BFN and BMN have the same relation between them as the two ALB and BGI, because, as the bases of these BA and BI are equal, so BN, which is the base of the triangle BFN , is equal to itself insofar as it is also the base of the triangle BMN: whence it evidently follows that, as BF is to NM, so AL, that of the sides of the triangle ALB which relates to BF in the triangle BFN, that is to say which is the subtension of the same angle, is at IG, that of the sides of the triangle BGI which relates to the side NM of the triangle BNM.

Then BF is to MN as BI is to NI, because the two triangles BIF and NIM, being right-angled and having the same angle towards I, are similar.

Moreover, if we draw HO parallel to NB, and extend IB to O, we will see that BI is to NI as OI is to HI, because the triangles BNI and OHI are similar.

Finally, the two angles HBG and GBI are equal by construction. HOB is equal to GBI and to OHB because the latter is equal to HBG.

Therefore the triangle HBO is isosceles; and the line OB being equal to HB, the entire OI is equal to DK, especially since the two sets HB and IB are equal to it. And so, to resume from the first to the last, AL is to IG as BF is to NM, and BF to NM as BI to NI, and BI to NI as OI to HI, and OI is equal to DK; therefore AL is to IG as DK is to HI.

So that if, to trace the ellipse DBK, we give to the lines DK and HI the proportion which we will have known by experience to be that which serves to measure the refraction of all the rays which pass obliquely from the air in some glass , or other transparent material that we want to use, and that we make a body of this glass which has the shape that this ellipse would describe if it moved circularly around the axle DK, the rays which will be parallel in the air to this axle, as AB, ab, entering this glass, will turn away there in such a way that they will all assemble at the burning point I, which of the two H and I is furthest from the place from which they are coming. For you know that the ray AB must be diverted to point B by the curved surface of the glass represented by the ellipse DBK, just as it would be by the flat surface of the same glass represented by the straight line CBE, in which it must go from B towards I, because AL and IG are to each other like DK and HI, that is to say as they must be to measure refraction. And the point B having been taken at discretion in the ellipse, all that we have demonstrated of this radius AB must be understood in the same way of all the other parallels to DK, which fall on the other points of this ellipse, so that they must all go to I.

Moreover, because all the rays which tend towards the center of a circle or a globe, falling perpendicularly on its surface, must suffer no refraction there, if from the center I[37] we make a circle at whatever distance one wishes, provided it passes between D and I, like BQB, the lines DB and QB, revolving around the axle DQ, will describe the figure of a glass, which will assemble in the air at the point I all the rays which will have been on the other side, also in the air, parallel to this axle, and reciprocally which will cause all those which will have come from point I to go parallel to the other side.

And if from the same center I[38] we describe the circle RO at such a distance as we want beyond the point D, and that having taken the point B in the ellipse at discretion, provided however that it is not no farther from D than from K, we draw the straight line BO, which tends towards I, the lines RO, OB and BD, moved circularly around the axle DR, will describe the figure of a glass which will make the rays parallel to this axle on the side of the ellipse will diverge here and there on the other side, as if they all came from point I; because it is obvious that, for example, the ray PB[39] must be diverted as much by the hollow surface of the glass DBA, as AB by the convex or bumpy of the glass DBK, and consequently that BO must be in the same straight line as BI, since PB is in the same straight line as BA, and so on with the others.

And if, again, in the ellipse DBK[40] we describe another smaller one,

Diopter figure 37.jpg

but of the same species, like dbk, whose burning point marked I is in the same place as that of the preceding one also, marked I, and the other h in the same straight line and towards the same side as DH, and having taken B at discretion, as before, we draw the straight line Bb, which tends towards I, the lines DB, Bb, bd, moved around the axle Dd, will describe the figure of a glass which will make all the rays which, before meeting it, will have been parallel, will again find themselves parallel after leaving it, and that with this they will be closer together and will occupy less space on the side of the smaller ellipse db than on that of the larger one. And if, to avoid the thickness of this glass DB bd, we describe from the center I the circles QB and ro, the surfaces DBQ and robd will represent the figures and the situation of two less thick glasses which will have in this the same effect.

Diopter figure 38.jpg And if we arrange the two similar glasses DBQ[41] and dbq unequal in size, so that their axles are in the same straight line, and their two external burning points marked I in the same place, and that their circular surfaces BQ , bq look at each other, they will also have the same effect in this.

Diopter figure 39.jpg And if we join these two similar glasses, unequal in size, DBQ[42] and dbg, or put them at such a distance as we want from each other, provided only that their axles are in the same line right, and that their elliptical surfaces look at each other, they will cause all the rays which come from the burning point of one marked I to go and assemble in the other also marked I.

And if we join the two different dbq[43] and DBOR in such a way that their surfaces DB and BD look at each other, they will cause the rays that will come from point i, that the ellipse of the glass dbq has for its burning point, s will move away as if they came from point I which is the burning point of the BDOR glass, or, reciprocally, that those which tend towards this point I will assemble at the other marked i.

And finally, if we join the two dbor[44] and DBOD always in such a way that their surfaces db, BD look at each other, we will cause the rays which, while crossing one of these glasses tend beyond towards I, to will separate again, coming out of the other, as if they came from the other point i. And we can make the distance of each of these points marked Ii larger or smaller, as much as we want, by changing the size of the ellipse on which it depends; so that, with the ellipse alone and the circular line, one can describe glasses which cause the rays which come from a point, or tend towards a point, or are parallel, to change from one to the other of these three sorts of dispositions in every way that can be imagined.

The hyperbola is also a curved line that mathematicians explain by the section of a cone, like the ellipse; but, in order to make you understand it better, I will introduce here a gardener who uses it to compose the embroidery of some parterre.

He again plants two stakes at points H[45] and I; and, having attached to the end of a long ruler the end of a somewhat shorter cord, he makes a round hole at the other end of this ruler in which he inserts the peg I, and a loop at the other end of this rope which he passes through post H; then, putting his finger at the point X where they are attached to each other, he runs it from there downwards to D, still holding however the string quite joined and as if glued against the ruler from point X to the place where he touches it, and with that quite stretched, by means of which, forcing this ruler to turn around the stake I as he lowers his finger, he describes on the earth the curved line XBD which is a part of a hyperbola; and after that, turning his ruler on the other side towards Y, he describes in the same way another part of it YD; and moreover, if he passes the loop of his rope through post I, and the end of his ruler through post H, he will describe another hyperbola SKT, quite similar and opposite to the preceding one.

But if, without changing his stakes or his ruler, he only makes his rope a little longer, he will describe a hyperbola of another kind, and if he makes it a little longer still, he will describe another one. of another species, until, making it completely equal to the ruler, he will describe a straight line at the link of a hyperbola; then, if he changes the distance of his stakes in the same proportion as the difference which is between the lengths of the ruler and the cord, he will describe hyperbolas which will all be of the same kind, but whose similar parts will be different in magnitude.

And finally, if he also increases the lengths of the cord and the straightedge, without changing either their difference or the distance of the two stakes, he will still only describe the same hyperbola, but he will describe a greater part of it; for this line is of such a nature that, although it always curves more and more towards the same side, it can nevertheless extend to infinity without its ends ever meeting: and thus you see that it has in many ways the same relation to the straight line as the ellipse to the circular; and you also see that there are an infinity of different kinds, and that in each kind there is an infinity whose like parts are different in size.

If, from a point like B, taken at discretion in one of them, we draw two straight lines towards the two points, like H and I, where the two stakes must be planted to describe it , and which we will again call the burning points, the difference of these two lines HB and IB will always be equal to the line DK, which marks the distance which is between the opposite hyperbolas: which appears from what BI is longer than BH especially since the ruler has been taken to be longer than the rope, and since DI is also all the longer than DH; because, if we shorten this one DI of KI, which is equal to DH, we will have DK for their difference. And, finally, you see that the hyperbolas that one describes, by always putting the same proportion between DK and HI, are all of the same species; then, besides that, you need to know that if, by the point B[46] taken at discretion in a hyperbola, we draw the straight line CE, which divides the angle HBI into two equal parts, the same CE will touch this hyperbola at this point B without intersecting it, of which geometers know the proof well enough.

But I want here next to show you that if, from this same point B, we draw towards the inside of the hyperbola the straight line BA parallel to DK, and if we also draw by the same point B the line LG which cuts CE at right angles, then, having taken BA equal to BI, that from the points A and I one draws on LG the two perpendiculars AL and IG, these two last AL and IG will have between them the same proportion as the two DK and HI. And then that, if we give the shape of this hyperbola to a body of glass in which the refractions are measured by the proportion which is between the lines DK and HI, it will cause all the rays which will be parallel to its axle in this glass will assemble outside at point I, at least if this glass is convex; and if it is concave, that they will deviate here and there, as if they came from this point I.

Which can thus be demonstrated: first, if we draw from point B the perpendicular line BF on KD prolonged as much as is necessary, and from point N, where LG and KD intersect, the perpendicular line NM on IB also prolonged, we will find that AL is at IG as BF is at NM; because, on the one hand, the triangles BFN and BLA are similar, because they are both right-angled, and because NF and BA being parallel, the angles FNB and LBA are equal; and, on the other hand, the triangles IGB and NMB are also similar, because they are right-angled, and the angles IBG and NBM are equal. And, besides this, as the same BN serves as the base of the two triangles BFN and NMB, so BA, the base of the triangle ALB, is equal to BI, the base of the triangle IGB; whence it follows that, as the sides of the triangle BFN are to those of the triangle NMB, so those of the triangle ALB are also to those of the triangle IBG. Then BF is to NM as BI is to NI, because the two triangles BIF and NIM, being right-angled and having the same angle towards I, are similar. Moreover, if we draw HO parallel to LG, we will see that BI is at NI as OI is at HI, because the triangles BNI and OHI are similar. Finally the two angles EBH and EBI being equal by construction, and HO which is parallel to LG, intersecting like it CE at right angles, the two triangles BEH and BEO are entirely equal. And so BH the base of the one, being equal to BO the base of the other, there remains OI for the difference which is between BH and BI, which we have said is equal to DK: so that AL is at IG as DK is at HI. Whence it follows that, always putting between the lines DK and HI the proportion which can serve to measure the refractions of glass, or any other material which one wishes to employ, as we have done to trace the ellipses, except that DK does not can be here only the shortest, instead of which it could previously only be the longest, if we draw a portion of hyperbola as large as we want, like DB[47], and that from B we bring down , at right angles to KD, the straight line BQ, the two lines DB and QB, turning around the axle DQ, will describe the figure of a glass, which will cause all the rays passing through it, and will be in the air parallel to this axle on the side of the flat surface BQ, in which, as you know, they will suffer no refraction, will assemble on the other side at point I.

And if, having traced the hyperbola db[48] similar to the preceding one, we draw the straight line ro in such place as we wish, provided that, without cutting this hyperbola, it falls perpendicularly on its axle dk; and if the two points b and o are joined by another straight line parallel to dk, the three lines ro, ob and bd, moved around the axle dk, will describe the shape of a glass which will make all the rays , which will be parallel to its axle on the side of its flat surface, will diverge here and there on the other side, as if they came from point I.

And if, having taken the shorter line hI[49] to draw the hyperbola of the glass robd than for that of the glass DBQ, we arrange these two glasses in such a way that their axles DQ, rd are in the same straight line, and their two burning points marked I in the same place, and that their two hyperbolic surfaces are looking at each other, they will cause all the rays which, before meeting them, will have been parallel to their axles, will still be so after having crossed them both, and with this will be constricted into a lesser space on the robd glass side than on the other.

Diopter figure 46 47.jpg And if we arrange the two similar glasses DBQ[50] and dbq of unequal size, so that their axles DQ, dq are also in the same straight line, and their two burning points marked I in the same place, and that their two surfaces hyperbolic look at each other, they will make, like the preceding ones, that the parallel rays on one side of their axle will also be parallel on the other, and with that will be tightened in less space on the side of the lesser glass.

And if we join the flat surfaces of these two glasses DBQ and dbq[51], or if we put them at such a distance as we want from each other, provided only that their flat surfaces look at each other without it is necessary with that that their axles are in the same straight line; or rather, if one composes another glass which has the shape of these two thus joined, one will make by its means that the rays which will come from one of the points marked I will go to assemble in the other on the other side .

And if we compose a glass which has the figure of the two DBQ and robd[52] so joined that their flat surfaces touch each other, we will cause the rays which will have come from one of the points I to deviate like s they had come from the other.

And finally, if we compose a glass which has the shape of two such as robd[53], once again so joined that their flat surfaces touch each other, we will cause the rays which, going to meet this glass, to be separated like to assemble at point I which is on the other side, will again be separated after having crossed it as if they had come from the other point i.

And all this is, it seems to me, so clear that one need only open one’s eyes and consider the figures to understand it.

Moreover, the same changes of these radii, which I have just explained, first by two elliptical glasses, and afterwards by two hyperbolic, can also be caused by two of which one is elliptical and the other hyperbolic. And moreover, one can still imagine an infinity of other glasses which do like these, that all the rays which come from a point, or tend towards a point, or are parallel, change exactly from one to the other of these three provisions. But I do not think I have any need to speak of them here, because I can explain them more conveniently hereafter in geometry, and those which I have described are the most suitable of all for my purpose, as well as I want to try now to prove, and show you by the same means, which of them are the most suitable for it, by making you consider all the principal things in which they differ.

The first is that the figures of some are much easier to trace than those of others: and it is certain that after the straight line, the circular and the parabola, which alone cannot suffice to trace any of these glasses, as well as everyone can easily see, if he examines it, there are none simpler than the ellipse and the hyperbola, so that the straight line being easier to trace than the circular, and the hyperbola being no less so than the ellipse, those whose figures are composed of hyperbolas and straight lines are the easiest to carve that can be; then then those whose figures are composed of ellipses and circles, so that all the others that I have not explained are less so.

The second is that among several which all change in the same way the arrangement of the rays which relate to a single point, or come parallel on a single side, those whose surfaces are the least curved or the least unequally, so that that they cause less unequal refractions, always change a little more exactly than the others the disposition of the rays which relate to other points or which come from other sides. But to understand this perfectly, it is necessary to consider that it is the only inequality of the curvature of the lines of which the figures of these glasses are composed, which prevents them from changing so exactly the disposition of the rays which relate to several different points. or come parallel from several different sides, that they make that of those that refer to a single point or come parallel from a single side.

Because, for example, if, in order to make all the rays which come from point A[54] assemble at point B, it was necessary that the glass GHIK, which was put between two, had its surfaces all flat, so that the straight line GH, which represents one of them, had the property of causing all these rays, coming from the point A, to become parallel in the glass, and by the same means, that the other straight line KI caused only there they went to assemble at point B, these same lines GH and KI would also cause all its rays, coming from point C, to assemble at point D; and generally that all those who would come from someone from the points of the straight line AC, which I suppose parallel to GH, would assemble in someone from the points of BD, which I also suppose parallel to KI, and as far away from it that AC is from GH: especially as these lines GH and KI not being curved at all, all the points of these other AC and BD relate to them in the same way as each other.

All the same, if it were the glass LMNO[55], of which I suppose the surfaces LMN and LON to be two equal portions of a sphere, which had the property of causing all the rays coming from point A to assemble at the point B, it would also have to cause those of the point C to assemble at the point D, and generally that all those of someone from the points of the surface CA, which I suppose to be a portion of a sphere, which has the same center that LMN, would assemble into one of those of BD, that I also suppose a portion of a sphere, which has the same center as LON, and is as far from it as AC is from LMN, especially since all the parts of these surfaces LMN and LON are also curved with respect to all the points which are in the surfaces CA and BD. But because there are no other lines in nature than the straight line and the circular, all the parts of which are related in the same way to several different points, and neither one nor the another cannot suffice to compose the figure of a glass, which causes all the rays which come from one point to assemble at another point exactly, it is obvious that none of those which are required there will only make all of them the rays which will come from some other points are assembled exactly at other points. And that, to choose those among them which can make these rays deviate the least from the places where one would like to assemble them, it is necessary to take the less curved and the less unequally curved, so that they approach the closest to the line or of the circular, and still rather of the line than of the circular, because the parts of the latter relate in the same way only to all the points which are equally distant from its center, and do not relate to no others in the same way as they do to this center; whence it is easy to conclude, that in this the hyperbola surpasses the ellipse, and that it is impossible to imagine glasses of any other figure which gather all the rays coming from various points into as many other points equally distant from them so exactly as that whose figure will be composed of hyperbolas. And even, without my stopping to give you a more exact demonstration of it here, you can easily apply this to other ways of changing the arrangement of rays which relate to various points or come parallel from various sides, and know that for all, either the hyperbolic glasses are there cleaner than any others, or at least that they are not notably less clean there, so that this cannot be counterbalanced with the ease of being cut, in which they surpass all others.

The third difference between these glasses is that some cause the rays which intersect while passing through them to be a little further apart on one of their sides than on the other, and others do just the opposite. As if the rays GG[56], are those which come from the center of the sun, and that II are those which come from the left side of its circumference, and KK those which come from the right, these rays deviate a little more from each other from the others, after crossing the hyperbolic glass DEF, than they did before: and, on the contrary, they deviate less after crossing the elliptical ABC[57], so that this elliptical makes the points LHM closer to each other than the hyperbolic does, and it even makes them all the closer the thicker it is; but, nevertheless, however thick it can be made, it can only make them about a quarter or a third closer than the hyperbolic: which is measured by the quantity of refractions which the glass causes ; so that the mountain crystal, in which they grow a little larger, must make this inequality a little larger. But there is no glass of any other figure imaginable that causes the LHM points to be noticeably farther apart than this hyperbolic nor less than this elliptical.

Diopter figure 54.jpg Now you can notice here on occasion in what sense it is necessary to understand what I said above, that the rays coming from various points, or parallel from various sides, all cross from the first surface which has the power to make that they come together in about as many other different points; as when I said that those of the object VXY[58], which form the RST image on the fundus of the eye, cross from the first of its BCD surfaces. Which depends on whether, for example, the three rays VCR, XCS and YCT truly intersect on this surface BCD at point C; where does it come from that even though VDR intersects with YBT much higher, and VBR with YDT much lower, however, for what they tend to the same points as VCR and YCT do, they can be considered all the same only if they also meet at the same place. And, because it is this BCD surface which makes them thus tend towards the same points, one must rather think that it is at the place where it is that they all intersect, that not higher nor lower; without even what the other surfaces, like 1,2,5 and 4,5,6, can deflect, prevent them.

No more than the two sticks ACD and BCE[59], which are curved, deviate a lot from points F and G, towards which they would go, if, crossing each other as much as they do at point C , with that they were straight; it does not cease to be truly at this point C that they intersect. But they could well be so bent that it would make them cross again in another place.

Diopter figure 46 47.jpg Picture 47

And, in the same way, the rays which cross the two convex glasses DBQ[60], and dbq cross on the surface of the first, then recross again on that of the other, at least those which come from various sides; because, for those who come from the same side, it is manifest that it is only at the burning point marked I that they cross.

The sunrays picked up by the elliptical glass ABC burns with more force than being picked up by the hyperbolic DEF.

For it is not only necessary to beware of the rays which come from the center of the sun, like GG, but also of all the others which, coming from other points of its surface, have not sensibly less force than those of the center; so that the violence of the heat which they can cause must be measured by the size of the body which assembles them, compared with that of the space where it assembles them: as if the diameter of glass ABC is four times greater than the distance which is between the points M and L, the rays picked up by this glass must have sixteen times more force than if they only passed through a flat glass which did not deflect them in any way.

And, because the distance which is between these points M and L is greater or less great, by reason of that which is between them and the glass ABC, or another such body which causes the grounds to assemble there without the magnitude of the diameter of this body can add nothing to it, nor its particular shape, only about a quarter or a third at the most, it is certain that this infinite burning line that some have imagined is only a reverie. And that having two burning glasses or mirrors, one of which is much larger than the other, however they may be, provided that their figures are all alike, the larger must well collect the rays of the sun by a larger space and further from oneself than the smallest; but that these rays should not have more force in each part of this space than in that where the smallest collects them; so that one can make extremely small glasses or mirrors which will burn with as much violence as the larger ones. And a fiery mirror, the diameter of which is not greater than about the hundredth part of the distance which is between it and the place where it is to collect the rays of the sun, that is to say, which has the same proportion with this distance which the diameter of the sun has with that which is between it and us, were it polished by an angel, cannot make that the rays which it assembles heat more in the place where it assembles them than those which come directly from the sun: which should also mean burning glasses in proportion. From which you can see that those who are only half learned in optics allow themselves to be persuaded of many things which are impossible, and that those mirrors which Archimedes was said to have burned ships from a great distance must be extremely great, or rather that they are fabulous.

The fourth difference which must be remarked between the glasses in question here belongs particularly to those which change the disposition of the rays which come from some point fairly near to them, and consists in that some, namely those whose surface which looks towards this point is the most hollow because of their size, can receive a greater quantity of these rays than the others, although their diameter is not greater. And in this the elliptical glass NOP[64], which I suppose so large that its extremities N and P are the points where the smallest diameter of the ellipse terminates, surpasses the hyperbolic QRS[65], although it is suppose also as large as one wishes; and it cannot be surpassed by those of any other figure.

Finally, these glasses differ further in that, to produce the same effects with respect to the rays which relate to a single point or to a single side, some must be more in number than the others, or must cause only the rays which refer to various points or to various sides intersect more times: as you have seen that to do with elliptical glasses that the rays which come from one point come together at another point, or deviate as if they came from another point, or that those which tend towards a point deviate again as if they came from another point, it is always necessary to employ two of them there, instead of there not only one must be used if hyperbolics are used; and that the parallel rays, remaining parallel, can be made to occupy less space than before, both by means of two convex hyperbolic glasses which cause the rays which come from different sides to cross each other twice, and by the means of a convex and a concave which means that they only intersect once. But it is obvious that you should never use several glasses for what can be done just as well with the help of one, nor make the rays cross several times when one is enough.

And generally we must conclude from all this that hyperbolic glasses and ellipticals are preferable to all the others that can be imagined, and even that hyperbolic glasses are almost in everything preferable to ellipticals. Then I will now say how it seems to me that each species of glasses should be composed to make them as perfect as possible.