How To Cut Glasses

Finally, the last and main thing that I would like to suggest is that one should exercise oneself in polishing the concave lenses of both sides for the eyeglasses that serve to see accessible objects, and that, having first exercised oneself in making those that render these eyeglasses quite short, because they will be the easiest, one should then try, gradually, to make those that render them longer, until one has reached the longest ones that can be used.

And so that the difficulty that you may find in constructing these last eyeglasses does not discourage you, I want to warn you that, although at first their use may not seem as attractive as that of the others that seem to promise to raise us to the heavens and show us stars and bodies as diverse and perhaps as particular as those that we see on earth, I still judge them to be much more useful, because through their means one will be able to see the various mixtures and arrangements of the small parts that animals and plants, and perhaps also the other bodies that surround us, are composed of, and from there derive much advantage for coming to know their nature.

For already, according to the opinion of several philosophers, all these bodies are made up of parts of the diverse elements mixed together in different ways.

I think that the nature and essence of all these inanimate bodies consist solely in their size, shape, arrangement, and movements of their parts…

The concave lenses that serve the longest eyeglasses need to be cut more precisely than the others. What is the main utility of the eyeglasses with a long focus?

After having chosen the glass or the crystal which one intends to use, it is first necessary to seek the proportion which, according to what has been said above, serves as a measure for its refractions, and one can conveniently find it by using such an instrument.

Diopter figure 61.jpg

Picture 61.

EFI[71] is a very flat and straight board or ruler, and made of any material you like, provided that it is neither too shiny nor transparent, so that the light shining on it can easily be discerned from it. The shadow. EA and FL are two pinnules, that is to say two small blades of such material as you like, provided it is not transparent, raised plumb on EFI, and in which there are two small holes circles A and L, placed just opposite each other, so that the ray AL, passing through, is parallel to the line EF; then RPQ is a piece of glass that you want to test, cut in the shape of a triangle, whose angle RQP is right, and PRQ is more acute than RPQ. The three sides RQ, QP and RP, are three faces all flat and polished, so that the face QP being pressed against the board EFI, and the other face QR against the pinnule FL, the ray of the sun which passes through the two holes A and L penetrate as far as B through the glass PQR without suffering any refraction there, because it meets its surface RQ perpendicularly; but, having reached point B, where it obliquely meets its other surface RP, it cannot leave it without bending towards some point of the board EF, as for example towards I. And all the use of this instrument does not consist in only to thus make the ray of the sun pass through these holes A and L, in order to know by this means the relation that the point I has, that is to say the center of the small oval of light that this ray described on the EFI board, with the two other points B and P, which are, B, that where the straight line which passes through the centers of the two holes A and L ends on the surface RP, and P, that where this surface RP and that of the EFI board are cut by the plane which we imagine passing through the points B and I, and together by the centers of the two holes A and L.

Now, knowing thus exactly these three points BPI[72], and consequently also the triangle which they determine, one must transfer this triangle with a compass onto paper or some other very smooth plane, then from the center B describe by the point P the circle NPT, and, having taken the arc NP equal to PT, draw the straight line BN, which intersects IP prolonged at the point H, then again from the center B by H describe the circle HO, which intersects BI at the point O, and we will have the proportion which is between the lines HI and OI for the common measure of all the refractions which can be caused by the difference which is between the air and the glass which we examine; of which if we are not yet certain, we can have other small right-angled triangles different from this one cut from the same glass, and using them in the same way to find this proportion, we will always find it similar, and thus we will have no occasion to doubt that it is truly the one we were looking for; that if, after that, in the straight line HI, we take MI equal to OI, and HD equal to DM, we will have D for the vertex, and H and I for the burning points of the hyperbola of which this glass must have the figure to serve for the spectacles I have described.

Diopter figure 62b.jpg Figure 62b. And we can make these three points HDI more or less distant than they are as much as we want, by simply drawing another straight line parallel to HI farther or closer than it from point B, and drawing from this point B three straight lines BH, RD, BI which intersect it; as you can see here that there is the same relationship between the HDI and hdi points as between the three hdi.

Diopter figure 42.jpg Picture 42. Then it is easy, having these three points, to trace the hyperbola in the way which has been explained above, namely, by planting two spades at the points H and I[73], and making the rope put around the pole H is so attached to the ruler that it cannot fall back towards I any further than as far as D.

But if you prefer to trace it with the ordinary compass, looking for several points through which it passes, put one of the points of this compass at point H and having opened it so much that its other point passes a little beyond the point D[74] as up to 1, from the center H describe the circle 133; then having done M 2 {\displaystyle {\rm {M2}}} equal to H 1 {\displaystyle {\rm {H1}}}, from center I, through point. 2, describe the circle 233, which intersects the preceding one at the points 33, through which this hyperbola must pass, as well as through the point D, which is its vertex. After all, put one of the points of the compass back to point H, and opening it so that its other point passes a little beyond point 1, as up to 4, from the center H describe the circle 466; then, having taken M 5 {\displaystyle {\rm {M5}}} equal to H 4 {\displaystyle {\rm {H4}}}, from the center I by 5, describe the circle 566, which intersects the previous one at the points 66 which are in the hyperbola; and thus, continuing to put the point of the compass at point H, and the rest as before, you can find as many points as you please of this hyperbole.

Diopter figure 64.jpg Which will perhaps not be bad for roughly making some model that roughly represents the shape of the glasses you want to cut. But, to give them exactly this figure, it is necessary to have some other invention by means of which one can describe hyperbolas all at once, as one describes circles with a compass; and I know of no better one than the following: first, from the center T[75], which is the middle of the line HI, we must describe the circle HVI, then from the point D raise a perpendicular on HI, which intersects this circle at point V; and from T, drawing a straight line through this point V, we will have the angle HTV, which is such that, if we imagine it turning in circles around the axle HT, the line TV will describe the area of a cone in which the section made by the plane VX parallel to this axle HT and on which DV falls at right angles, will be a hyperbola quite similar and equal to the preceding one. And all the other planes parallel to this one will also cut in this cone hyperbolas all similar, but unequal, and which will have their burning points more or less distant, according as these planes are from this axle.

Then of what one can make such a machine. AB[76] is a lathe or roller of wood or metal which, rotating on poles 1, 2, represents the axle HI of the other figure. CG, EF are two blades or boards all flat and united principally on the side which they touch each other, so that the surface which can be imagined between them, being parallel to the roll AB, and cut at right angles by the plane that we imagine passing through the points 1, 2 and C, O, G, represents the plane VX which intersects the cone. And NP, the width of the upper CG, is equal to the diameter of the glass you want to cut, or a little bit larger. Finally KLM is a ruler which, rotating with the roller AB on poles 1, 2, so that the angle ALM always remains equal to HTV, represents the line TV which describes the cone. And you have to think that this ruler has passed through this roller so much that it can rise and fall while flowing into the hole L, which is precisely its size; and even that there is somewhere, as towards K, a weight or spring which always presses it against the blade CG, by which it is supported and prevented from passing over. And moreover, that its extremity M is a point of well hardened steel which has the force to cut this blade CG, but not the other EF which is below; whence it is manifest that, if we move this KLM rule on poles 1, 2, so that the steel point M passes from N by O to P, and reciprocally from P by O to N, it will divide this film CG into two others, CNOP and GNOP, whose NOP side will end with a sharp line, convex in CNOP and concave in GNOP, which will have exactly the shape of a hyperbola; and these two blades CNOP, GNOP, being of steel or other very hard material, could be used not only as models, but perhaps also as tools or instruments for cutting certain wheels, from which I will say later that the glasses must draw their figures. However, there is still here some defect in that the steel point M being a little differently turned when it is towards N or towards P than when it is towards O, the edge or the edge which it gives to these tools cannot be equal everywhere. Which makes me believe that it will be better to use the following machine, notwithstanding that it is a little more composed.

ABKLM[77] is only one piece which moves entirely on poles 1, 2, and whose ABK part can have any shape you want; but KLM must have that of a ruler or other such body whose lines which terminate its surfaces are parallel; and it must be so inclined that the straight line 4 5, which we imagine passing through the center of its thickness, being extended to that which we imagine passing through the poles 1, 2, makes an angle 234 equal to the one that was earlier marked with the letters HTV. CG, EF are two boards parallel to the axle 12, and whose surfaces which look at each other are very flat and smooth, and cut at right angles by the plane 12 GOC; but, instead of touching each other as before, they are here precisely as far apart as is necessary to give passage between them to a cylinder or QR roller, which is exactly round and everywhere of equal size; and, moreover, they each have a NOP slot, which is so long and so wide that the KLM ruler, passing through it, can move here and there on poles 1, 2 as much as is necessary to trace between these two plates part of a hyperbole the size of the diameter of the glasses that you want to cut.

Diopter figure 67.jpg Picture 67.

And this rule is also passed through the QR roller in such a way that, making it move with itself on poles 1, 2, it nevertheless remains always enclosed between the two boards CG, EF, and parallel to axle 12. Finally , Y 67 {\displaystyle {\rm {Y67}}} and Z 89 {\displaystyle {\rm {Z89}}} are the tools that must be used to carve any body in hyperbole, and their YZ handles are of such thickness that their surfaces, which are all flat, touch exactly on both sides. on the other, those of the two plates CG, EF without their allowing them to slip between the two, because they are very polished; and they each have a round hole 5, 5 in which one of the ends of the QR roller is so enclosed, that this roller can turn well around the straight line 55, which is like its axle, without causing them to turn with itself , because their flat surfaces being engaged between the boards prevent them; but that in some other way that he moves he constrains them to move also with him. And from all this it is manifest that, while the KLM ruler is pushed from N to O and from O to P, or from P to O and from O to N, causing the QR roller to move with itself, it causes the QR roller to move with itself. means these tools Y 67 {\displaystyle {\rm {Y67}}} and Z 89 {\displaystyle {\rm {Z89}}} in such a way that the particular motion of each of their parts describes exactly the same hyperbola as does the intersection of the two lines 34 and 55, one of which, namely 34, by its movement describes the cone, and the other, 55, describes the plane which intersects it. As for the points or cutting edges of these tools, they can be made in various ways, according to the various uses for which they are to be employed.

And to give the figure to the convex glasses it seems to me that it will be good to first use the tool Y 67 {\displaystyle {\rm {Y67}}}, and cut several steel blades almost similar to CNOP[78], which was described earlier; then, both by means of these blades and of the tool Z 89 {\displaystyle {\rm {Z89}}}, to hollow out a wheel like d all around according to its thickness abc, so that all the sections that can be imagined therein are made by planes in which lies ee, the axle of this wheel, have the shape of the hyperbola traced by this machine; and finally to attach the glass which one wishes to cut on a lathe, like hik, and apply it against this wheel d in such a way that, making this lathe move on its axle hk by pulling the cord ll, and this wheel also on his by turning it, the glass placed between two, takes exactly the shape that one must give him.

However, concerning the way of using the tool Y 67 {\displaystyle {\rm {Y67}}}, it should be noted that only half of the cnop blades should be cut at once, for example, that which is between points n and o; and for this purpose it is necessary to put a bar in the machine towards P which prevents the KLM ruler, being moved from N towards O, from being able to advance towards P except as far as is necessary to make line 34, which marks the middle of its thickness, reaches as far as the plane 12 G O VS {\displaystyle {\rm {12GOC}}}, imagine cutting the boards at right angles. And the iron of this tool Y 67 {\displaystyle {\rm {Y67}}} must be of such figure that all parts of its cutting edge are in this same plane when line 34 is there; and that there are no others elsewhere which advance beyond towards the side marked P, but that the whole slope of its thickness throws itself towards N. For the rest, it can be done if foam or as acute, and as much or as little inclined, and as long as you like, depending on what you think best. Then, having forged the cnop blades, and having given them with the file the closest shape possible to that which they must have, they must be applied and pressed against this tool. K 67 {\displaystyle {\rm {K67}}}, and moving the KLM ruler from N to O, and vice versa from O to N, one of their halves will be cut; then, in order to be able to make the other all alike, there must be a bar or some other such thing which prevents them from being advanced towards this tool beyond the place where they are when their NO half is completed from prune; and then, having moved them back a little, it is necessary to change the iron of this tool Y 67 {\displaystyle {\rm {Y67}}}, and put another one in its place whose cutting edge is exactly in the same plane and of the same shape, and as advanced as the preceding one, but which has all the slope of its thickness thrown towards P, so that, if we applied these two flat irons against each other, their two cutting edges seemed to form but one. Then, having transferred to N the bar that had previously been placed towards P to prevent the movement of the KLM ruler, this ruler must be moved from O to P and from P to O until the cnop blades are as advanced to the tool Y 67 {\displaystyle {\rm {Y67}}} than before, and that being so, they will be finished cutting.

For the wheel d, which must be of some very hard material, after having given it with the file the figure closest to that which it must have that one could have, it will be very easy to finish it, first with the blades cnop, provided that they were at the beginning so well forged that quenching has since deprived them of nothing of their shape, and that they are applied to this wheel in such a way that their cutting edge nop and its axle ee are in the same plane, and finally that there is a spring or counterweight which presses them against it while it is made to turn on its axle. Then also with the tool Z 89 {\displaystyle {\rm {Z89}}}, whose iron must be equally cut on both sides; and with that it can have any quasi shape you want, provided that all the parts of its cutting edge 89 are in a plane which intersects the surfaces of the CGEF boards at right angles. And to use it, you have to move the KLM ruler on poles 1, 2, so that it passes immediately from P to N, then vice versa from N to P, while you rotate the wheel on its axle. By means of which the cutting edge of this tool will remove all the inequalities which will be found from one side to the other in the thickness of this wheel, and its point all those which will be found from top to bottom: for it must have a cutting edge and a point.

After this wheel has thus acquired all the perfection it can have, the glass can easily be cut by the two various movements of it and of the lathe on which it is to be attached, provided only that there be some spring or another invention which, without impeding the movement that the turn gives it, always presses it against the wheel, and that the bottom of this wheel is always immersed in a vase which contains the sandstone, or the emery, or the tripoli, or the hotpot, or other such material which it is necessary to use for cutting and polishing glass.

And following the example of this you can quite understand in what way one must give shape to concave glasses, namely by first making blades like cnop with the tool Z 89 {\displaystyle {\rm {Z89}}}, then cutting a wheel both with these blades and with the tool Y 67 {\displaystyle {\rm {Y67}}}, and everything else in the way just explained. Only it must be observed that the wheel used for the convex can be as large as one wishes, but that the one used for the concave must be so small that, when its center is opposite - screw of line 55 of the machine that is used to cut it, its circumference does not pass above line 12 of the same machine. And we must make this wheel move much faster than the lathe to polish these concave glasses, whereas it is better for the convex ones to make the lathe move more quickly; the reason for which is that the movement of the lathe wears the ends of the glass much more than the middle, and that on the contrary that of the wheel wears them less. As for the utility of these various movements, it is very manifest; for, polishing the glasses with the hand into a form in the manner which alone has hitherto been in use, it would be impossible to do anything good except by chance, though the forms were all perfect; and polishing them with the single movement of the lathe on a model, all the small defects of this model would mark whole circles on the glass.

I do not add here the demonstrations of several things that belong to geometry, because those who are a little versed in this science will be able to understand them enough for themselves, and I am convinced that the others will be more happy to tell me. to believe than to have the trouble of reading them. For the rest, so that everything is done in order, I would like first of all to practice polishing glasses, flat on one side and convex on the other, which had the shape of a hyperbola whose burning points were two or three feet apart: for this length is sufficient for a telescope which serves to see inaccessible objects quite perfectly. Then I would like us to make concave glasses of various shapes, always hollowing them out more and more until we have found by experience the right shape of the one who would make this bezel as perfect as possible and better proportioned to the eye that would have to use it. Because you know that these glasses must be a little more concave for those who are short-sighted than for others. Now, having thus found this concave glass, especially as the same can be used for the same eye for any other kind of glasses, there is no longer any need for glasses which are used to see inaccessible objects, but to practice to make other convex lenses which must be placed further from the concave than the first, and to make some also by degrees which must be placed further and further up to the greatest distance possible, and which are also larger in proportion. But note that, as much as these convex glasses must be placed further from the concaves and consequently also from the eye, all the more so must they be cut more exactly, because the same defects there divert the rays of farther from where they need to go.

Diopter figure 67b.jpg

As if the glass F[79] deflects the ray CF as much as the glass E deflects AE, so that the angles AEG and CFH are equal, it is clear that CF, going towards H, goes much further from the point D where it would go without that, which AE does from point B going towards G. Finally, the last and main thing I would like us to practice is to polish the convex glasses on both sides for the glasses which are used to see accessible objects, and that, having first practiced making those which make these glasses very short, because they will be the easiest, one should then try, by degrees, to make those which make them longer, until you have reached the longest you can use. And, lest the difficulty you may find in the construction of these last spectacles disgusts you, I want to warn you that even though their use does not at first attract so much as that of those others who seem to promise to elevate us in the heavens, and to show us there on the stars bodies as particular and perhaps as diverse as those which we see on the earth, I nevertheless judge them much more useful, because we will be able to see by means of them the various mixtures and arrangements of the small parts of which the animals and the plants, and perhaps also the other bodies which surround us, are composed, and from this to draw much advantage to come to the knowledge of their nature: because already, according to the opinion of several philosophers, all these bodies are made only of parts of the elements variously mixed together; and, according to mine, their whole nature and essence, at least of those which are inanimate, consists only in the size, shape, arrangement, and motions of their parts.

For the difficulty that is encountered, when vaulting or hollowing these glasses on both sides, in making the vertices of the two hyperbolas directly opposite each other, we can remedy this by rounding off their circumference, and making it exactly equal to that of the handles to which they must be attached for polishing; then, when they are fastened thereto, and the plaster or pitch and cement with which they are joined thereto is still fresh and flexible, passing them with these handles through a ring into which they hardly enter .

I am not speaking to you of several other peculiarities which must be observed in cutting them, nor also of several other things which I have just said are required in the construction of spectacles, for there are none which I consider so difficult that she can stop the good spirits. And I don’t follow the ordinary reach of craftsmen; but I want to hope that the inventions that I have put in this treatise will be considered beautiful enough and important enough to oblige some of the most curious and industrious of our century to undertake their execution.