THE GEOMETRY OF Landscapes and Paintings Made Familiar and Easy: Useful to Limners in Drawing, and Gentlemen in Choosing PICTURES; And Beneficial to Architects and Carvers in Proportioning the Graces and Statues of their Buildings to the due Distance of Sight, and to Country Gentlemen in the more convenient Framing of their Platforms for Seats, and Prospects. In a LETTER to a FRIEND. He that denies Demonstrations, reflects on his own Power of reasoning, and not on the Truths he encounters. Demonstrations prove themselves, the Work therefore about Demonstrations is not to prove but illustrate them. Quo minime reris gurgite pisces erit. LONDON: Printed for Richard Baldwin, in the Old-Baily, 1690. SIR, I Have here sent you, as in a Letter, the whole body of the Geometry of Landscapes and Paintings; and such as it is, I freely tender it to your perusal: this I must say for it, That it is not, as is too usual in Mathematical Treatises, obscured with the Formalities of Pedantry, and rather a Knot of useless Problems, than a Solid Instruction, for I have made it my particular purpose, to make it pertinent, hearty, and brief; I have aimed at the Marrow of my Subject, and not at little trifling Ornaments: I shall say no more, but if you can bear with the Roughness of the Style, incident to Demonstration, I do not doubt but I shall give you all the Satisfaction my Subject requires; and so I dismiss you. THE GEOMETRY OF Landscapes and Paintings Made Familiar and Easy. CAP. I. Of the Geometry of Painting and Landscapes in general. THE Geometry of Painting is rather Optic or Perspective, than real; for if a Man were to be seen at that distance, as to appear as little as in a common Landscape, 'twere impossible with common sight to discern any part of him distinctly, much less any feature about him: So likewise of the Leaves of Trees, and Ornaments of Buildings; for at such a diminishing distance they would all appear obscure and confused, and not distinct and complete, indeed, just as if they were going to leave the verge of sight, unless reduced back again by Perspective. A Landscape is therefore rather a neat contraction or Epitome of things visible than a real view of them, and upon that account in such Pictures we admit the nicest lines and features; thus though a Man represented in a Landscape be six Foot high (in common intendment) yet there though he contain but the length of six Inches, and little better than half an Inches breadth in his Face, his delineaments are as proportionate and distinct, as if the sight were at six Foot proportion; but yet even in such a Landscape, if a Man were to be represented at any distance, there lies a skill too to imitate the Natural darkness and obscureness of distance, as before was required to show the distinct Figure of the Man represented near. The same may be said of the Pictures of Houses and Trees, etc. CAP. II. Of the Sky. BUT that you may the better enter into the apprehension of the Nature of Landscapes, you must note, that as every Landscape contains just two demy-semy-circles, or one half Circle complete, to wit, one Spherical and Concave of the Sky, and the other in Plano of the Earth; so all Landscapes are to be ruled by the measures of the Circle, to wit, by Tangents, in laying down the surface of the Earth in Plano, and by Sines, in deciphering the Sky overhead. Now, as Landscapes are thus ruled in all parts by the Circle, so is every part about a Landscape even demonstrable; hence it is that by a Landscape you may set off the Altitude of the Sun, the four Quarters of the World, the Sun's greatest Declination, and the Latitude of the place; so as by a Dial too you may set off the hour of the day, and except it be about the time of the Solstices, almost the day of the Month when the Picture was made: But as these are Niceties too speculative for ordinary Pictures, so I shall leave them to be traced by those that think it more worth their time and leisure; for my part, I think if the Sky be rude drawn as Clouds often make it appear, 'tis as well if not pleasanter to the Eye, than a clear Prospect; nay, and if a clear Prospect be drawn, if the Limner commit not gross Absurdities, as in a Landscape of Snow, to give the Sun too great Altitude, or in the Landscape of a particular place, to set off the Points of the Compass falsely. If, I say, he commit not such faults as these, 'tis no matter almost where he sets the Sun, for in some Sign or other the Sun visits all Quarters; indeed he must beware of setting him too near the Zenith, without he intends to draw a Prospect of some place under the Lines: But of this enough. CAP. III. Of the Earth praecognita. THE next part of a Landscape is the setting off the Earth in Plano; but before I enter on that, I shall lay down, for the common Readers Instruction, these few Praecognita. 1. All Measures about the World and other round Bodies are performed by the Circle; and as all circular Bodies, whether greater or less, have the same Nature of the Circle; so altering the proportions of several Circles, the common Circular Relations or Measures follow the several Cirles in their several proportions; thus for instance. Every Circle is divided into 360 equal parts, called Degrees; so that whether by reason of the quantity of the Circle each of those parts be an Inch or a Mile, 'tis all one in relation to the Circular Measure, for if each part be a Mile, than the whole is 360 Miles round; and if each part be an Inch, the whole will be 360 Inches round, the quarter of each 90, and the Semi-diameter, or the distance from the Centre to the Circumference as long as 60 Degrees of it; that is, as two thirds of the 90, or Quadrant, etc. So that from hence it is, that by a Quadrant (that is a quarter of a Circle) though it be little in compass, we take the proportion even of the Heavens, a Body, though Circular, yet almost infinitely larger. As for Example. diagram Explanation. Just as the little Quadrant at (a) measures the great Quadrant at (b) so the Heavens are measured by a Quadrant below upon the Earth; and as the Compasses divide the little Quadrant (a) at (d) (h) into equal parts, so the Eye by the Quadrant (a) divides the Quadrant (b) or any great Circular Body into (c) (e) that is, the same equal and proportional parts with it, and so were the same to be divided even into a thousand parts, or were it a whole Circle, or half, or a Quadrant, you see 'twould be all one, the greater Circle must needs be commanded and divided justly by the less. So further the less Circle commands the greater in all other proportions likewise; thus as the distance (a) (h) is equal to the Semidiameter (a) (f) of the lesser Circle, so the distance (e) (b) in the greater Circle is equal to his Semidiameter (b) (f) but of this enough. 2. Besides this, you must know that all Circles at 45 Degrees, that is, at half of 90, or the Quadrant naturally set off a square Angle answering to their Semidiameter, so that by the Quadrant we cannot only measure the Circle, but a Square; nay, by the other Degrees we are able to set off any Triangle or other Figure whatsoever. For Example. diagram Explanation. You see that the Line of 45 Degrees (c) (d) as well sets off the points of Squaring in the greater Circle (b) as in the lesser (a) and so it would do in any Circle of any Magnitude whatever. As for setting of Triangles by the Circle, it does not relate to my present purpose, and therefore I shall pass it by without more writing. 3. Lastly, You must know, That where one straight Line cuts another, be it at any Angle whatever, it will make equal Angles on both sides, thus: The Angles (a) (b) are equal, so the Angles (c) (d) are equal, and not only so, but the cross-line at (a) runs an equal length from over against the point (e) to cut (i) as it does afterwards to come over against the point (f) The like may be said of the Cross-line (c) so that if you know how much a cross-line verges from a straight in a yard by the Rules of ordinary Proportions, you may know how much it will verge even at 10000 Miles end. diagram CAP. IU. Of a Landscape on a Plain. THESE Praecognita being thus laid down and explained, I shall now be able with the greater ease and freedom to discover to my Reader the due proportions of Distances and Magnitudes in such Landscapes, and to that purpose I shall first describe one of the Proportion I before hinted at in the first Chapter, to wit, where an Inch represents a Foot, that is, where a Man of six Foot shall be represented by six Inches; so where a Tree two Foot broad, shall be deciphered by two Inches, and of forty Foot high by 40 Inches. The Plain where the Sight is lost to all distinction. diagram The Explanation of this Figure. SECT. I. By this Figure I suppose a Man's Foot to be placed at (b) and his Eye at (a) just six Foot distance, which is the Quadrant or part of a Circle a Man makes upon his standing in Prospect upon the Earth. Now in such case six Foot distance from him makes up half the Quadrant, to wit, 45 degrees, as I before shown in my Praecognita, and so 12 Foot takes up half the Line: (c) (d) in its Section according to the third Rule of Angles in my Praecognita, and so 18 Foot but a third part, etc. and so on to 576 and 1152, after which there can be no distinction, but the sight is lost in the level of the Plain: indeed if there happen to be a Man walking, or a Hill at a greater distance upon the Plain, the sight is again renewed according to the Altitude of what appears upon the Plain, and shall be discernible by the Rule of Optics, till it come to be a Minute in the Circle of Sight, as shall be explained anon. Object. But you will say 1152 Foot, which is less than a quarter of a Mile, is but a little way to discern upon a Plain: 'Tis true, and yet further we cannot see any thing distinctly on it, except, as I have said, something appear that may be raised above the Plain: thus whilst I stand on the Shore I may see a Ship upon the Sea at some League's distance, but if I stand even with the Water, I cannot see a Plank float though never so broad, as I have said; a quarter of a Mile off indeed were there any Island a Mile broad in such a case and exactly levelly to the Water, a weak tangent might possibly touch mine eye at a distance with a dimmer ray, but more would be imperceptible. SECT. II. The Demonstration of the Figure. What I have said before in my Praecognita makes the Demonstration of this Figure plain; for the line (c) (d) being just six Foot distant from the line of standing (a) (b) as must be because (d) is the incident point of 45 Degrees upon the Plain, as I have shown before: In my Praecognita it will necessarily, I say, follow from hence, that the sight at 45 Degrees cannot but set off six Foot; so likewise if the sight at the line of line of standing (a) (b) upon its going six Foot distance from the said line cut, the line (c) (d) in the middle 'tis necessary by the third Rule in my Praecognita, that the line should go full as much again, or six Foot more from the line of standing (a) (b) before it can reach the Plain (b) (d), the same reason applied Demonstrates the rest. The Demonstration of the line of Tangents in Plano is evident, for if sight be by right lines, as it is, and the rays of sight at six Foot, and twelve, etc. before demonstrated, be true, as they cannot but be, than the line in Plano cannot but be true likewise, because it is no other than the proportion of sight intercepted in right Lines before it reaches the Points of Demonstration 6 and 12, etc. To conclude, I have Deciphered your Tangents both by the parts of the Square and the degrees of the Circle, so that you may be able to understand how to use either upon occasion; in taking of a Prospect, as for your Line of Tangents in Plano, you may draw it bigger or less, to the proportion of whatsoever Picture you please, by a Square accordingly; but I think this matter is explained sufficiently. CAP. V A Landscape on the Prospect Exalted, and not a Plain. BY what has been said it appears that if at any time you are to Idea and Draw out a large Prospect with variety of Sights, you must either suppose yourself looking on rising ground, and then you may see as far as any Object is capable of being seen by the Rules of Optics, as shall be shown hereafter, or else you must suppose yourself placed as on a Hill, and then the Altitude of the Hill will as it were become yours, and increase your sight accordingly; and so supposing yourself on an Hill but of forty Rods high, if a piece of flat Wood, or any other body of an agreeable size for sight, were in the Plain of the Sea, you might discern it near a League off in a clear day. Example: But because I will tell you how to take a Prospect from an Hill, I shall add you this other Figure: The Plain. diagram Explanation: As for the Demonstration of this Figure, it needs it not, because it depends wholly on the same grounds as the precedent, only there is this little difference, that at present you are measuring by a Circle and Square of a vastly larger size than what you were before; however, as I told you in my Praecognita, the same Ratio of Measure of this Circle continues with what did before, all the difference is that as before you reckoned the Measures of your standing by six Foot, your height only, so now imagining yourself on an Hill (which you may show on the side of your Landscape if you please) you must add the Hill's height to it. Thus in the precedent Example you have the Eye at (c) upon a Hill of a quarter of a Mile, and with the little Quadrant (a) setting off the greater (b), now I have taken this Hill to be a quarter of a Mile high for Example, but you may either imagine it any other height at pleasure, or else if you are to take any real Prospect, find it by your Quadrant only by looking at any known Villages about you; but this you must remember, that in real measuring of Distance, you must allow a little for the descent of the Hill, because the point (b) at the bottom of the Hill is but imaginary, and in the bowels of the Earth, for it is not as in the first Figure where the Line of standing is wholly in your personal length, for here is the Hill added, and as much as the Line (c) (d) is longer than the Line of Plain (c) (b) so much is to be added to whatever Descent is set off for the Descent of the Hill, that is, about 25 Rod, in our present Case a little more than a ¼ of a ¼ of a Mile, nor is more to be added whether the Tangent or Distance set off be further or nearer, whether one Mile or twenty, because the Descent is but one, be the Distance what it will. As for the reason of setting off my Scale of Miles, why at 45 Degrees it should give ¼ of a Mile, it depends on this, by my Praecognita I shown you, that a Quadrant at 45 Degrees necessary sets off a Square, and consequently if the Line of standing (c) (b) be a quarter of a Mile, and the Line of Plain (d) (b) be as straight and a Square equal to it, it must needs contain a ¼ of a Mile's distance likewise, but indeed if the Descent of the Hill be not complete to the Plain at 45 Degrees, 'tis otherwise, and allowance is to be made accordingly, as may be gathered from the Rules of Tangents following: 1. the Line of Tangents is intercepted before it comes to the real Plain (d) (b) it makes as it were a new Plain, pro hac vice. 2. You are to Idea the length of your Line of Standing not otherwise then as shall answer the Descent of your Tangent, but if your Tangent after finds liberty to descend with freedom to the real Plain, than you may take your full Line of Standing likewise, thus if the Hill at (f) intercept the Tangent (c) (d) you are to reckon your Line of Standing no longer then from (c) to (e), but because the Hill (f) does not intercept your sight at the Tangent (g), therefore for that Tangent you are to take the real Plain (d) (b). Further, as for the Line (g) the Tangent at ½ a Mile's distance, it's reason is because it cuts in half the Parallel Perpendicular Line (d) which answers to the Line of Standing in the Square (c) (b), for this was demonstrated by the Lines of Secancy before, that if one Line cut another it will make an equal Verge on both sides, if it be straight; so that if the Secant (g) has gone but ¼ of a Mile from the Line of Standing, and yet has cut the Parallel in ½ 'tis Plain in a ●/4 of a Mile more, that is in ½ a Mile the Secant (g) will reach the Plain (b) (d). So I set off the Tangent (h) to be three Miles, the reason is the Secant (h) cuts the Parallel at the ¼ of a Mile's distance, but at a 1/12 part so that it will go 11 times further, or 12 quarters of Miles, or three whole Miles in all before it will reach the Plain as aforesaid (b) (d), and thus by the same reason you may set off all the rest, and many other proportions of Tangents, if you please, for the Parallel Line (d) which is incident of course to the Angle found at (d) by the 45 Degrees, as I have before shown, is as a continual and demonstrable Measure to all the Tangents that shall pass through it towards the Plain (b) (d), be they of any kind whatever. CAP. VI Of the Circularity of Tangents. BEsides what has been already said, you ought to observe, that in every Landscape or Picture, there ought to be a Centre of Sight round with all your Tangents ought to fall as naturally as the Circumference to the Centre; now this Centre of Sight you may either make imaginary out of the Picture, and so at what distance you please, or else really in it, and that either in the midst of the Foot for a Building or in one Corner, where being to describe a Vale, you cast your Tangents to the other. For Example: You take your Compasses, and putting them to your Line of Tangents in Plano, you take the several distances and set them off one after another upon the Centre you pitch upon, thus in the present Figure in a Plain, where Hills nor Buildings distract the Sight, the Eye at (6) seeing every way alike, sets off his Prospect like as in these Lines, thus suppose (5) to show a quarter of a Mile, (4) half a Mile, and so for the rest, or more or less lines of Tangents at your pleasure. diagram CAP. VII. Of the Proportions of Buildings, Hills and Vales. HAving already given you the general Rules of Prospects both on Plains and where the Sight is raised, I shall now proceed to show how Buildings, Hills, and Trees, aught to be set off, and of these first for Buildings, as for Trees the same Rules will serve for them as for Buildings, and therefore I shall not particularly trouble myself about them, but leave them as Corollaries to my Readers apprehension; I know right well as to express enough helps the Understanding, so to speak or write much and needless confuses and confounds it. SECT. I. Of Buildings. Were you to set off the height of any extraordinary Building in a Piece, and would you do it exactly, suppose it were Salisbury Steeple, which is 400 Foot high, or there about, your Method is first to decipher the place of its standing in your Piece, and how many Degrees, or Parts, or Length such a Building in Plano would take up at that distance, and by that you are to erect your Altitude; thus suppose you would draw it a Furlong from the Centre of Sight, and it be a quarter of a Furlong high, as it is at least, and a Furlong by the Rules I have before shown you, take up eight Inches in your Landscape, you must draw the Steeple two Inches high, and that is the just proportion, or exact enough; for as I may say so you set off his height by the proportion it holds to the Circle of the Tangent where it stands, by the Rules of the Circle laid down in my Praecognita. So if you are to draw an Arch of Building of a considerable length, or some long Wall, if in such case you take the two Altitudes of its ends at their several lengths, then by drawing parallel Lines between their Tops and Basisses, you find the just proportion of their Diminution and Increase to your sight, even from one end to the other, be it never so long. So if you are to draw a Wall half a Mile long, and at half a Mile's distanee, you are then to imagine the Centre of Sight, that is, where your Eye stands a Centre, and so you are to reckon from thence to the Tangent of half a Mile 60 Degrees, when you have done this, you may easily decipher any proportion of the length of such a Wall, for as if it were half a Mile and round, 'twould take up just 60 Degrees, that is, be as long in the Circumference as from the Centre of Sight to the Tangent of Distance, so as 'tis straight if you set it off by the Line of Tangents in a Plano, agreeable to that distance you have your desire, and that must be done, because as I have before shown you, the Tangents fall in Circular Lines round the Centre of Sight, so that if a strait Line be put it must needs cross several little Tangents. SECT. II. Of Statues and Pictures set as Distance. Further, if you are to draw a Picture, or frame a Statue, if you make it for a Plain, you ought as I may say, to take the proportion of the Life, but if it is to stand aloft or at some distance from the sight, then both in your Picture and Statue you ought to exceed the Life, and you ought to frame your proportions in both by a Line of Tangents of suitable size to the distance, the Statue or Picture is to be placed from the sight; and in this case your Line of Tangents will be of excellent advantage to you for proportions, will as it were naturally and of course follow them from the ordinary size to whatever you shall please to direct and be like a Scale as it were to weigh out other Measures from them. Thus suppose a Picture placed at six Foot distance on a Plain, take up five Degrees in the Circle of Sight, for so any body about six Inches broad will do then, I say if you would remove this Picture to stand 60 Foot high, but to new draw it and to make it large enough, to appear at that distance as distinct as before, at six Feet distance, you must make it of a proportion to take up five Degrees in the face as before, that is, you must make the face of five Foot over, and the rest of the body proportional, and then as all things of five Degrees are equally apprehended by the Sight, so that Statue at sixty Foot distance shall be as distinct featured to the sight, as that at six. But because such a Statue would be monstrously large, and of vast weight, therefore let us suppose that a face were set six Foot from us, and were carved or drawn within the compass of an Inch, or a little better, that is, near as big again as is mentioned in the first Chapter; I say, such a Face might be visible enough, and would take up but one Degree in the Circle of Sight; therefore a Statue that takes up one Degree in the Sight, may do pretty well too at 60 Foot distance from the Sight, now as one Foot is a Degree in 60 Foot, so if such a Statue were made a Foot over in the Face, it might do well enough, that is, as well at 60 Foot as the Statue of an Inch, and a little better will do at six Foot; now as I have thus shown the proportions in this case, so with a little labour this will lead you too any other whatever, so that what I have written in this matter, I think is sufficient, and I shall only make this Reflection on it, that I do not doubt but by this means by the help of Fire to hold Correspondence with a Friend in a Siege, nay, and the intermediate Enemy not having so much as a jealousy of the matter, shall have his sight quite taken off by a due Tangent of Scaffolding. SECT. III. Of Hills and Vales. I shall now proceed to the Tangents of Hills and Dales, and show how they are to be deciphered, and in this case there is but very little difference between the Tangents of Buildings and Hills, indeed a Building is best set off in a direct prospect, as being more full to the sight, but a Valley in an obligne one, because you may be able to take the more room, and make your Tangents the larger to express the greater and more lively variety in it, but if your Cloth be large enough you may make your Valley straight or how you please; as for the Plain height and depth of your Hill or Valley, the common use of the Line of Tangents will show you how to raise or depress it as you shall see occasion at any distance, or to any height whatever you may apply it in the same manner as you did to Buildings. To conclude, I know nothing further to acquaint you with, that may pertinently Instruct you further in this Matter, unless it be, that you always cast your Shadows at one and the same Angle, and that according to the Elevation of the Sun in your Landscape; for tho' this may not answer rightly to your Sun in the Landscape, as it is deciphered in Plano, yet to the truth, and the Sun in the Heaven Shades are at the same Angles, at any one Prospect whatever. CAP. VIII. Of the length of Prospect, and what things are out of Sight. BUt there is a proportion how far things may be seen as well as how big they will appear; thus sometimes they will be lost to the sight, and thus we cannot see the Edge of a Knife or the Point of a Needle, and yet in a Prospective they shall appear gross and scragged bodies; Now the Rule of this Proportion is, or at least very near it, whenever any body through its distance or smallness becomes so little as to take up but a Minute, or the 60th part of a Degree in the Circle of Sight, from that instant it is quite lost and disappears. Thus even the Sun, were he removed proportionably further in the Circle of Sight, so as to become as a Minute, he would be invisible, or very near so; and thus it is, that the lesser Stars which are as it were Minutes to the extended Sight in the Heaven, appear and are seen as it were with a force to our Eyes, and yet not but that there's some difference between Bodies luminous and obscure, for indeed a luminous Body as a Beacon ought to be reckoned by its extent of Rays, and not barely by its bulk. Thus it is very difficult without Glasses to see an Inch 3600 Inches, or an 100 Yards off, where it becomes a Minute, but if a Candle were placed at that distance, which possibly should not possess so great an extent in mere Flame, yet as it both contracts the Sight to it, and spreads in rays, it may be seen near ten times as far; so were a body of any colour of a Yard square set 3600 Yards off, that is, so as to become a Minute, which is but little better than a Mile off, 'tis impossible well to see it with the sight without Prospectives, but if a flame of less extent were made in a dark night, it would be seen near ten times as far, by the Rules afore delivered, whereas a Pan of Coals in the night, or a Fire in the day, wanting that power of diffusing rays and contracting the sight, are capable of been seen no further than a Body of that size, by the Rules of Perspective, as aforesaid. But to proceed from hence, it is, that is, from proportional Diminution, that if you stand at an agreeable distance you shall see the Sign when the Post that bears it is disappeared; so you shall see the Vane when the Stem of the Weathercock is disappeared, and yet a Post or Bar of Iron of an Inch square in such a case from its length shall receive double distance of sight from what it would have, that is, if an Inch square disappears as the sight being scattered at an 100 Yards, yet if an Inch square be a Yard long, it shall be seen near 200 Yards; that is, to 30 Seconds, because the length in such a case recollects the sight, and continues it at least to another Diminution; but the sight in such case that shall see it while possesses only 15 Seconds, that is, when it is 400 Yards off, must be a kind of Wonder or Miracle, without it receive help by Glasses: But yet the Lead in Glass-windows seems to be visible ptoportionably much further in this case, for though it be but about a quarter or a third part of an Inch broad, it may be seen better than an 100 Yards; but indeed there is a Fallacy in this case, for it is not so much the Lead that we see in such case, but we see as it were little stops made to the luminous Waves of Light made through the various Reflections of the Glass; and this must needs be seen even as far as the Glass itself can be seen distinctly. By the same Rules were a Building like the Monument situate to the best advantage on a Hill, you might easily see it till it came to be a Minute in the Circle of Sight, and after that, with some difficulty, till it comes to be 30 Seconds, by the Rules last passed; that is, because it has length greater than its breadth; then thus, suppose, I say, the Monument were a Rod in the diameter or breadth at top, that is, five Yards and a Foot, as I conceive it may be there abouts; then I say the Monument will take up a Degree in the Quadrant of Sight at a Furlong and a half's distance, 30 Minutes at a qr. ½ qr. of a Mile, 15 Minutes at 3 qr. of a Mile, 7½ at a Mile and a ½, 3¾ at 3 Mile, about 2 Minutes at 6 Miles distance, and about one Minute at 11 Miles distance, after that at 22 Miles it will become as 30 Seconds to the Sight, which is as great a distance as the Eye is ordinarily able to reach and discern it at. Nota, Thus Astronomers solve the Rationality of the appearrnce of Mars to the Earth. Thus too you may know how far even the Graces of such a Building may be seen; so how far the Features of a Man's Face, or a Statue; for instance, if an Eye take up an Inch, it may be just discerned at an 100 Yards distance, but that so faintly that were it not the proportion of the Body did direct us to it, we could not perceive; so were there a Ring of a Foot square round the Monument, it might be seen 1200 Yards, that is, 3600 Foot, or between a quarter and half a Mile, and after that, it will not be discernible at all; but I shall say no more of this nature, for I am afraid I have explained this matter so much already, that I shall be guilty of a Repetition as bad as Tautology. But as by what has been said, it appears that things may be lost to the sight by their remoteness from it, so on the other side, if an Object be of an agreeable quantity for sight, we may discern it, though at any distance whatever, for thus we see the Heaven and Stars, and yet this is only where the sight has a free Plain, as in the Heavens, for were there an Hill ten Miles high, and proportionably large, by the Rule of Sight it would be seen at least 36000 Miles off, but in such case the very circularity of the Earth long before that space would cloud it from the sight; thus I believe the very Teneriffe is a little obscured, for were the Earth truly plain, and that Mountain not too picked to be lost to the sight for want of breadth, it would not only with ease be seen an 100 Miles, for at that distance 'twould fill a Degree in the Quadrant, but even 60 times as far, even till it come to a Minute, as I have before demonstrated; but as that Hill reaches at least the Region of the Clouds, so the least Cloud in all that prospect would obscure it. But to illustrate this the more by Example, suppose an Hill a Mile high, whereas the Teneriffe is reputed near two in the Perpendicular; but I say, suppose an Hill only one Mile high in the Perpendicular, and we were out at Sea from it in a clear day, I say such an Hill at 60 mile's distance would at least take up one Degree in the Quadrant, by the proportions at first laid down in my Praecognita, and if so, it might be well seen near 60 times further; so if it were half a mile high, 'twould appear half a Degree, if a quarter 15 Minutes, if a Furlong high, about seven half Minutes; but as I have said, when it is come to a Minute or , that is, not to be above the 4th part of a Furlong, or 20 Rod only in height, from that instant, as I have said, it becomes invisible, without, as I formerly acquainted you, it be of an agreeable length to attract the sight. But there is another reason that hinders long Prospects besides the Circularity of the Earth, to wit, the grossness of our Atmosphere, the impurity of the Air; thus in large Prospects, as at the Alps the Pike of Teneriffe and other Hills, if you see towards them at a great distance, suppose sixty Miles or better, if the Sun do not sit as it were purposely to illuminate you, and the Air be not extremely clear, you cannot see so much as the least appearance of them; indeed the Air between will as it were fill and satisfy the Eye with a Sky-blew; so that though the Atmosphere may in some measure advance the sight, in causing the Hill to increase and look larger some minutes than really it is, yet as with all it obstructs the sight to so great a degree, 'tis not to be expected, that any wonderful Prospect is capable of being achieved on this Level of the Earth. CAP. IX. Of your Landscapes by Perspective and Glasses. HAving thus far proceeded on the Delineating of plain Prospects, I shall now speak a little to those that may be taken by Perspective, and then conclude. Know then there is no more difficulty to set off a Prospect by Perspective, be it of any kind whatever, than there is in setting it off by the Sight, for having looked with your Perspective on two or three Objects, and by your observation found its measure of increase to your sight, you may as justly draw your Tangents to answer your Prospective, I mean in a remote Prospect, as you can to answer your sight in a near one; and afterwards, if you reduce such a larger Line of Tangents by perspective to your lesser and more true one, you may make your Landscape to express the Figures of Prospect more perfect and lively than any Eye by any possibility can reach too. From whence you see that the difference between a Landscape taken by sight and Prospective, is only this, that a Perspective-glass makes the Circle of Sight appear larger than it is, which is impossible to be delineated in Plano, except in a Landscape only of part of a Prospect, for indeed as the Eye is the only true measure of Proportion, which gives every body its true place according to the Ratio of a Circle; so whatever exceeds that, is erroneous; and yet on the other side, the Perspective may help the Eye, as I have said, when reduced, for where the Eye can by no possibility particularise or distinguish nicely the parts of a Prospect, the Perspective reduced may clear it to your desires. I might add here too, that light that is cast through reflection may reach to a vast extent, as we see by the Moon; but that the light through refraction even of the Sun, extends but to fifteen Degrees, as we see plainly, 'tis dark an hour after Sun set in the Wintertime, by which means you may see how to order the light of Windows and Staircases; I might hereto likewise add to what Positions of Sight and Light a Picture best answers, as how to draw Pictures in Lines of Refractions which can only be seen by Glasses; as also how to cast Species against a Wall, Glasses or any solid body whereby you may imitate the appearance of Spirits in Glasses, or indeed of any Apparition in a Room, if you please: but what shall I say, these are the Curiosiora Mathematicae; and tho' they are not unworthy the conusance of a Philosopher, as he is sometimes led by the knowledge of things, useless to those that are of benefit, yet as to practice, they are rather of evil consequence and diversion than any real subserviency. CAP. X. CONCLUSION. SIR, I Am now drawn to my Conclusion, and as I conceive, this that I have delivered you is all the Geometry that is useful either for Landscapes or Statues; this I can assure you, 'tis all I can readily think of, and this I can say from my own Conscience to you, I have left no doubt that I knew in my mind unsolved in it, so that I hope 'tis moderately perfect, or if it is not so, I wish it so; I have taken some pains in it, for I was not willing you should take a Picture drawn by my mind at the first setting, and yet I hope it is easy too, for whilst I have laboured to be short, I have with all endeavoured from being obscure. You see by what I have shown you, that there is not a Grace-line or Shade in a Picture, Landscape or Statue, that is not governed by a Demonstration; you see how the Limner and Carver do but Essay by guess at the more perfect Lines of Optics, and tho' it might be true what is generally urged, that to tie the Fancy too nicely to Rules, confounds and destroys it; yet 'tis as true, that to leave the Fancy boundless of proportion, is as barbarous and ignorant; and besides, would Men consider aright, these Proportions should be set off at the first draught, and then they rather help the Fancy that follows in the colours than mar it; nor will this be with difficulty, for with a print of Chalk, or a little Lettuce of small Wires, the first proportions may be set of without the least trouble. I might have amplified what I have written to a Volume, but I love shortness: I could have traced many other curious Points, as how to know the distance of an Hill by the sight, how to raise a Statue, that to the whole City of London shall show the exact Features of a Man at once; nay, not only so, but I am persuaded without Romancing, I could order such a Pile on Southwark-side, that should imitate the Features of any particular Person, and that not without some Life, and to be discerned distinctly very near at any of the Stairs on London-side; but these are Curiosities too remote from use or practice; and as I never loved to encourage Flea-chaines and Air-bottles, so I now beg leave to rest Yours, etc. FINIS. ERRATA. PAge 4 line 12 for Cirles read Circles; p. 13 l. 31 necessary r. necessarily; p. 15 l. 12 with r. which; p. 24 l. 8 about seven half Minutes r. seven Minutes and an half. Besides this, the Reader is desired to take notice of the false Stops, of which he may see an example p. 18 l. 12, which in some places disguise the sense, but with a little of his care may easily be amended.