I offer, Monsig. Reurendiss, for ancient custom to address the works, which come to light, to some personage of value, act ˜ for the nobility and clarity of his, or true even for the intelligence of that matter, which in the book is about, to defend him from the evils of various languages. Hora has given me the opportunity to reduce the architecture of M. Serbastiano Serlio, (of whom I think it is better to be silent than to say little) and to adorn it with beautiful figures: which I have done willingly, not sparing any effort or expense, to satisfy the virtuosos. I have been thinking about it, at the expense, to satisfy the virtuous. I have been thinking about it, where not to indirittarla where not: for the ciche V.S. alone, it seems to me (and this my judgement and still universal) that she is mournful of all those beautiful qualities of soul, that to true Lord they agree. I leave to speak of her nobility, clear to adopt one, for them born in this city, of so many illustrious families. I keep silent that I am well instructed in such a science, that one truly ambushes her; many of me, with the study of imitating her, are inferior to her of great length. Of which Vitruvius is a good witness, restored by her as we see today, with so much pleasure, and the taste of literate men, with such beautiful and useful notes: but here ends her praiseworthy study, that all day long, not to give it to herself, nor to her complication, she always strives to find honoured and new things. This summer value of hers is very well recognized by this most illustrious lordship, to be the bearer of five or faithfully the mother of God, or to devolve it, when she is her shepherd, and Patriarch of the church of Aquileia: honour is given only to people, or illustrious for her majors or clear for herself stars: two things that are of obscure value, such as the splendour of majors. La onde seio, antico devotete sua, et che giagran pezza sini stato con esso meco considering, come le peteno scoprire questa mia intense servirsi, ho bravanti forse troppo a detto d'interromperla da sua alti pensieri, con queste mie rezore, et mal composte parole: mi scusa appresso lei l ardente assentitone, et reverenza, cheio I porto le le, disuguale in vero a mereriti sue, ma pero tale, che maggiore in me non pu˜ nascere. With which, making an end, I kiss her honoured and virtuous hand: begging her to accept with joy the petiole gift of her faithful servant before her, and to me she will give soul to the day to work things of greater importance in her service, which I satisfy her. N. S. Diola conserri, Di Venetioa a XXV. Di Maggio. M D LXVI
By V.S. Reverend
Humilservitore Francesco de Fraceschi, Sienese.
First Book of Geometry, by Serlio Sebastiano Serlio Bolognese
First of all, period is an indivisible thing, which has no ire if it starts with cynicism.
Line and a straight line et contino must be imagined from one point to another, in length down width.
Parallel, they are two continuous lines of equal distance.
Surface and of two equidistant dilated ferratas, i.e. und thing, which has length, and width without depth and can still be surface of different and equal sides.
Right angle will be, when a line perpendicular to the plumb banks called a castbera caress over a flat line.
And when said line will fall over a flat line more on one side, that on the other evening a sharp angel and a dull one the angle accused will be less than the rectum and the angle of his own will be greater than the rectum, which can be said strong angle team and above square.
Flat pyramidal angle, it will be two lines of equal length, together removing the part above and enlarged by the par below and this will be an acute angel.
Equilateral triangle, that is three equal sides will be three lines of equal length joined together and this figure blush three acute angles.
Triangle of two equal sides, will be two lines of equal length, i.e. one flat, und lead, and another major line which will be the triangle and this will be a right angle and two acute.
Triangle of three unequal three unequal length lines will conspire together and this figure hars ire sharp angles.
Quadrangle of unequal sides, it will be four lines of unequal length and this figure coats two orused angles and two acute angles and sometimes it will lapse a right angle.
Rhombus will be four lines of equal length, of which you can make a perfect square: one in this shape will be two acute angles, and two orruse and this figure takes its name from a fish that is said to be Rome and still you can say almond by lapel shape almond.
The shape of different and unequal sides will be of different lines in continuous length together, and even if it is of seven sides and all the angles are obtuse, it may well be a figure of more and sides so arranged that in them will be of the right angles, of the high and the obtuse and similar figures may come into the hands of the architect in different on which I will give the rule in the extreme of this book to give them back in the form of a perfect square.
Flat surface care line bin gola will be of two curved lines that is circular which figure will serve many things in this book and of which we will get the right rule, that is the team and this figure and treats the shape of those modern arches, which are called third acute, which in many buildings are seen, doors, arches and left.
Of the perfect circle you will have the center, the circumference and the diameter.
Half a circle in which the lead line is found to fall above the diameter, from which the right angle and the half-wings of the year are born.
Perfect square will be four lines of equal length congi feels together and will be four corners rested.
Educated that he will be the architect in the cognition of the past figures, he will need more other procedures, that is, to know how to increase, decrease and barrirle them proportionally and an imperfect form to reduce it to its perfection and to that value that it was imperfect and its first form.
First of all, the perfect fourth one is doubled like this and to do, given a perfect Quadro closed by four lines. A, R, C, D, evening drawn a line from the corner A, to the corner d, which will be the lair of the major square doubled to the minor, which sardines A, E, F, D, and the test and this. If the minor square contains in itself two triangles of equal value, follow that l major and doubled to minor, as in the marginal figures G, H, can be seen and measured.
The doubling of the circle will be so, that given the minor circle in a perfect square closed by four lines A, B, C, D and outside of the one drawn a minor circle would enter a square C, B, E, F doubled to the minor square, as I have shown further back, marked that the major circle, and doubled to the minor, as is more understandable in two circles K, L, and from here and drawn the overhang, i.e. the sporio of the base Tuscany described by Vitruvius: And even where it deals with sodamenti that are doubled, for the work that one on top of it because of the overhang, that has vessino a proserˆ above the firm.
But the architect will proceed further ahead, that is to reduce the triangular figures to quadrangulars and finally to perfect squares, which will give them the way to reverse. First give an equilateral triangle A, B, c, be divided by means of the happy B, C and by the angel A, at point E, draw a line and so the triangle will be started by means. Let that part of triangle A, E, C be given to part A, D, B, lassando quadrangular surface A, D, E, B.
In another way you can divide the triangle and reduce it to a quadrangular surface. The triangle will be A, B, C is divided the side A, B, into two equal parts and the side A, C, the same and is drawn a line D, E, as long as the line B, C and tightened them two sides by the bands, that is D, B and E, C which will be two triangles of equal value, one will be D, F, B, the other will be G, E, C, these will be two equal to the two upper triangles I, H, so the two triangles J, H the surface D, E, B, C, will be of the value that was triangle A, B, C.
Given a triangle of two equal sides, the haughtiest side is divided every one of the two sides into two equal parts, and from the opposite angle a line is drawn, so the triangle will be divided into two triangles of whatever shape it wants: the example of this can be seen in figure P, Q, R.
The same triangle P, Q, R can be reduced to a quadrangular surface.
If we have made two equal parts P, Q and the same of the line P, R, we have drawn a line of the same length as the one from below Q, R, which will be S, T, after we have drawn a line a, lead from T, R, which will be V, T, R, which will be of such a value that the one above P, S, V, the one above and the one from below will be a surface S, T, Q, R of the same value that was the triangle P, Q, R.
Given a triangle of three unequal sides A, B, C in the above manner it can be reduced to a quadrilateral surface. Let side A, B be divided by half and so side A, C, which will be F, G and drawn a continuous transverse line of such length as the line die below B, C and closed by the sides will make two triangles, triangle G. E. C will be equal to the upper triangle K, and triangle D, F, B will be equal to the upper I. Soothed then two triangles J, K, the surface D, E, B, C will be of the value, which was triangle A, B, C.
And because sometimes, by accident, it will happen to divide transversally, that is to say, by transverse ton triangle, one that is however of two equal sides: it will be asrepio grace a pyramidal triangle, like this one who is behind it: the way to divide it in two equal parts by transverse will be this one. Let there be a perfect square, the side of which is one of the sides of the triangle, and found the center of the square by placing all the sixths at the cones of the triangle, and the other point at the center of the square and pulling the circle towards the triangle above the two sides there will be the terms to divide it pyramidal triangle, whoever denies it, reduces the two parts on the surface and then they surface in a perfect square, as I will give this rule and find the truth.
Another difficulty could be that the Architect may need to be outside the given rules. It will be by accident a triangular shaped ground of unequal sides, and in one of the sides it will be a source, that is a well, but not in the middle of it side where it will be necessary to divide the ground into two equal parts, and that each unimpeded part of the other can enjoy it source, will be the triangle A, B, C, and the source will be G. Let there be mentioned a line of points occult from G to A, and divided the line B to C into two equal parts, which shall be D, and from D to another occult line drawn, which in the true division of the triangle is the triangle, but this is not the intention: it is therefore necessary to draw from D to E an occult line which shall be parallel to the line A to G. If, therefore, from the source to the E, an evident line is drawn, that will be the right division; and that the denial as I have said above, reduce the two parts to a quadrangular and then square surface and find the true as I shall later give the rule.
I have shown more clearly behind the square and the circle I say on the surface and also the way of dividing different triangles, but it will be better for the architect to go further, that is to know how to increase the perfect square, what part will be the design, still know how to increase proportionately whatever you are with this rule. It will be a perfect square A, B. C, D, which one would like to make an example of thanks to the value of such a square and three quarters, but that the perfect square first if he adds behind it three quarters more, which will be E, F, and so A, E, C, G will be a square and three quarters, but to reduce it to a perfect square if he adds behind it a square, like the first, which will be E, F, G, H and from A, F is drawn half a circle and continued the line D, E, until the half circle: from Raml half a circle will be the side of the perfect painting, which was first the surface of the painting and three-quarters, his test and this one. Let all those figures be surrounded by four lines which will be Q, R, S, T, as here lower is shown: et from the angle S, to the angle r, let a line be drawn: certain thing and, that the whole square will be divided by half equally. And as Euclid says, if equal, we will raise equal parts, the remaining parts will be equal. Soothed therefore the triangle K, L, and the triangle M, N, which are equal in themselves, the perfect square P will be equal to the surface or, and with this rule we can increase the square in which part we want, and reduce it always to the perfect square. Which rule the architect must lapel very familiar, for the different things that may happen to him.
And just as I have given the rule behind here to reduce any surface area, quadrangle it into a perfect square, so on the contrary I will give you the way of a perfect square to make it a quadrilateral surface. Given a perfect square A, B, C, D, as much as I`d like, here the surface is wide, you`ll drop a line from D to E, then pull the top line, the middle line, and the bottom line continue at the same distance E, from C, drop a plumb line as the line D, E, which will be E, F, and from corner F to corner D, a continuous line shall be drawn up to the line above, and where it shall cross the two lines, which shall be G, a plumb line shall be drawn up to the line below, which shall be H, I say that the surface D, E, I, H, shall be equal to the square A, B, C, D, the proof is this. (15) If we tighten the square and the surface G, from four lines, that is the square K and the surface L, then let the whole figure be divided by a sebiancio line and the triangle M, N, which are equal, and the triangle O, P, which are equal in itself, the surface L will be equal to the square, as shown in the figure below G, A, H, F.
The Architect could come to the maxi a shape of different and unequal sides, where it will be necessary to reduce it into a quadrangular shape, or rather into a perfect picture, in order to know the value of it to appreciate it, as if it were a fair partition when it was more people, Or was it earth, or any other matter, and of this the land surveyor, that is, the land surveyor without land, may serve him, even though he did not burp Arithmetic I have and numbers; and he that shall have this rule in his hands shall not be in the range of the dressmakers' headings: for he shall always measure and reduce every strong cloth into a quadrangular shape. I say that any similar or different form, or of more or less sides, which first makes them square, or a quadrangular form of all right angles of such size, how much he can extract from it figure, and appreciative of the rest, he can extract another quadrangular form from it, even if of right angles it will be good, how much not, he can extract so many triangles which will reduce them into a quadrangular form, as I have given the rule, and all of them will be forms of marked in isolation. First the major one after the alterations from hand to hand with its characters one by one: but the form that will be dealt with in the present will be of the fate demonstrated below, though as I have said there may be more forms.
It will be, for example, a figure of several unequal sides and angles, as I have said in the previous paper, and, as we have seen on the back of it, and in order to reduce it to a quadrangular form, first we shall take from it that major form of four right angles which I know will be A, B, C, D, and its sign L, and then we shall take from it another quadrangular form which will be E, F, G, H. Give the figure A, B, C, D; and above it may be the surface E, F, G, H, net so that here is a demi of our own in the second figure HERE AIDE, AND FROM THE ANGLE G, TO THE ANGLE, I, A LINE OF LEAD, which will leave it outside: a particle of the major figure L, which will be A, C. If afterwards the upper line, the middle line, and the lower line continue in length, then from angle I, to angle H, a continuous line is drawn, and where this line intersects the upper line, which will be K, a plumb line is dropped until the lower line, which will be M. I say that the square B, L, D, M will be equal to the surface above will follow M, for the reasons that more to the rear I have shown, and so of the two figures L, M, will be made a quadrilateral surface, the corners of which will be L, A, M, C, as here the rear is seen, which will be O, R, P, Q, it will be able to be placed over the large surface in the same way that can be seen here behind in the lower figure, with the above rule, and so the surface that was above will be added to the greater surface, fashionable, that the three figures L, M, N, will be reduced into a surface. A, S, T, C, to which all the triangles can be added with the same rule, and then, with the rule that I have shown further back, it will be possible to reduce their surface into a perfect square, and so every shape, strange as it may be, can be reduced into a perfect square, but that they are not curved lines, and even if curved lines will be there, it will be good for the man to go to the sign with diligence: But he shall not be able to measure it perfectly: for my parer and this, a curved line, cannot be bought by a straight line: and if it be, you shall find the quadrature of the circle, which has made and knows how to sweat so many wits pilgrims to find it.
Given a line or a rod or something else, whether it is desired, which is an unequal match and another thing of greater length will happen, and whether it is an unequal match and another thing of greater length will happen, and whether it is an unequal match and another thing of greater length will happen, and whether it is an unequal match and another thing of greater length will happen, Both the two chiefs shall drop two lines equally spaced, above the same line equal to the upper ones, after the major line has been drawn transversely, that is, joined by one chief to the other, and all the pure lines which are the minor lines, Th e rule will not only be left to the architect for more things, as I shall demonstrate to the cuna, but many ingenious creators will benefit greatly from carrying their works from small to large in proportion.
They will be by way of example several houses of different widths, the face of which will be smaller than the back towards the gardens: Which houses will be, but by some fire, or by wars so ruined that only a few vestiges of the permit will have been left on the face of my family, and we shall see the four of them a, b, c, d, and, since they are of more people, the four of them will have been besieged to other parts of the halos that, as I have said in the front part, so that each one knows his part of the vestiges of the face, but the rear boundaries are not seen except in the two corners a, b. May the architect presume in this damn thing that the line a, b, is the major line, and that the given part c, d, is the minor line. And with the rule, which I have shown in the past, he will give ˆ ciese uno his part, as shown in the figures below.
Sometimes the architect will want to increase a frame, i.e. a small one, to make a bigger one proportionally, with all its members: with the past rule it will be possible to increase how much they like it, and how much the frame will have to be bigger than the other one, is the more at the waterfront the line B, C, as shown below.
And similarly it will happen to the architect to make a fluted column, or martial, or in drawing, it will be small, transforming it into a greater form, so that he can make use of the above rule, and it is good that this column is Doric, this is meant of all the other menicres of columns, and not only this rule will serve for those three propositions, but for many things, that I will make a book of this rule if I want to demonstrate all of them. But in order not to be protracted, I'll leave it to the architect to investigate.
All those things, which move away from our sold, all the more they diminish, that having space consumes our sight, and yet that thing which will be far away, as far away as it is of the same greatness as the propinque, wanting that the distant ones all represent a greatness, it will be necessary to make use of art: for the reason that if the architect, or if he wants to, is low in one height, it will be one thing one above the other, that they all represent the same greatness, so that those from above, as well as those from below, and those in the middle, that all correspond to his due distance: first elected of the place, it is a column, a tower, a wall, whatever house that happens to adorn it, windows, statues, letters, whether it is greased; he will first elect one of the most convenient distance to look at the thing, and first at eye level, the eye is the center, and pulled the fourth part of a circle is then in the pariate where they do things done at said eye level, is drawn a line at that level, and greasy line, is done that thing you will want to do, and that size you want them to represent all the others. Then from the top of the thing a line is drawn to the center of the eye, and where it will intersect the line above the circular line, and from the center are drawn the lines that pass above the circle in equal equals, and from the center are drawn the lines that pass above the circle, and go to hurt in the said level, and those divisions go growing year by year, so that at this distance, in the opinion of the year, there is the same size, and from this rule we can measure the heights using numbers.
Among the quadrangular forms I find the more perfect a square, and the more the quadrangular form deviates from the perfect picture, the more it loses its perfection, even though it is surrounded by the same line, which was the square: examples grace will be a square of right angles surrounded by four lines, and each line sari ten, so that the line surrounding it will be x x x x. It shall be another quadrangle surrounded by the same line, the length of which shall be x y, and the width of which shall be v, and nevertheless the perfect picture multiplied in itself shall be one hundred, and the quadrangle shall be a macmque sect, because multiplied the sides of the perfect picture we shall say ten times ten, one hundred: and multiplied the sides of the quadrilung, we shall say five times fifteen, seventy-five, as is shown below.
And the more will be the above, a perfect picture of the value of a hundred, and it will be a quadrangular form longer than the first, that is, long avjii, and wide ij, which will be two times eighteen, thirty-six, and two times two, four, which are forty, and nevertheless multiplicated of its sides we shall say, two times eighteen, thirty-six; And therefore he sees what strength the perfect bodies of the perfect men have, and so does man, that the nearer he comes with his intellect to God, who is the same perfection, content in himself with more goodness, and the farther he departs from it God, delighting in earthly things, he loses more than that first goodness given to him. The example of this demonstration can be seen figuratively below, and this proposition will be of great benefit to the architect, in suddenly knowing that it differs both from one form to another about the value, and not even to the architect, but to the merchants who weather things buy so much by eye, and many other things, which I leave to the industrious to investigate them.
Given three points placed at random even if it is not above a straight line, the way to pass it over the bereavement three that I compass will be this: Let a direct line be drawn from 1. to 2. Let another line be drawn doing the same, and where two lines intersect it there will be the centre of three points, and they shall be pure places in whatever way you wish.
A surface of circular lines, and from 2 to 3, another similar one, and at its corners two continuous lines are drawn, and where they will join, there will be the center of three points, as shown below.
But from those things which seem to be a game, the architect shall nevertheless draw some fruit, and in many a damnation he shall serve them, and at the highest price of any roundness, by the petiole that it is, to know with the above rule to find its centre, and to know its dimeter and circumferentia by doing so, which hereunder is drawn.
It is found in the ancient, and also in the modern part of many columns, that in the lower part of the extreme are broken somewhere, and this is the only thing that in putting them in place its bases, but that they were not well flattened as a team, and well joined with them base, it is really that putting them on top of the bases could not lead them to the first one, but loading more on one side than on the other, that part more oppressed by the taken is resentful, and in its hem is broken: But if the architect knows the strength of the line helped by geometry, he can keep it this way, that the column in his foot is curved, i.e. column, as here on the side is shown in the first column, and so that its base is curved, i.e. colona, as here on the other side is shown in the first column, and so that its base is as concave as the curvature of it column: so, that placed the column is plumb above its base, from its base it will find its place without giving passion to its edge, nor to the base its curvature, and the concavity so it is to do, that placed a compass point above the top of the column at the A, and the other point in the part below the B, and circling with it compass to the C, will be the curvature, with which the concavity will still be made, and the same way it will be able to keep its capital in place, as can be seen in the other column here ˆ canto.
The architect will want to make a bridge, or an arch, or really a time of less height than the half circle, come that many bricklayers have a certain practice, that with the thread they make similar vaults, which really will correspond to the eye, and still fits with an oval shape made with the compass. Nevertheless, the architect will want to proceed with a strong sense of reason, he will keep this one of mine.
Assuming the width of the arch, which you will want to do, and I will find the means, I will make a perfect half circle, and how much you will want: For the height of the said acre, let there be made a half circle less than that hardness, then let the old major be divided into equal parts, and let them all be drawn in the centre, and let them fall to the plumb line, and where the lines that intersected the minor circle in the centre are made of points, And from them points to the plumb lines shall be drawn straight lines beginning above and coming down, and where they shall touch the plumb lines, points shall be made of them, and so from one point to another of the plumb lines a curved line shall be drawn, which can be done with the compass, but with the deserts, and praxica do pull mas example of this, it is seen here pit.
Considering the rule shown to me in the past paper, it came to my mind to make different forms of vases with it rule, brought with it from the region, and from the lines, nor did I have much trouble in describing the way, because the ingenious architect seeing the figure, here on the side, will be able to use it rule, making different forms. But this is sufficient for him, that as much as the vase is big enough in the greater body, a smaller circle is made inside the greater one, and with the central lines, and the transversal ones, making them then plumb, the body of the vase will be formed, and so will the neck, and the foot, to the judicious man's approval.
Et se vase lapel to be of more shapely body. Let the circle of greater half circle be made, that is to say, in that size which has to be amasio. First will be the lines that go to the center, then the transversal ones, and women will intersect the lines, that go to the center let the plumb line drop over the transverse 3, and the circle 4. Let the line be dropped above transverse 4, and from circle 5. Let the line fall over the transverse 5, and where they shall intersect all the plumb lines above the transverse lines, there shall be terms there to form the body of the vessel: and from the line 1. In that part of the perfect circle shall be the neck, and the lid of it vessel: the handles and the foot shall be at liberty of the wise, and so shall the other ornaments.
It is a beautiful thing to study with a compass the straight and curved lines, because you can find such a breath of things that man has never had to think about them, as I did tonight, that looking for a rule to make the shape of natural man, with more brevity than that of Alberto Durer, a man of truly great and subtle ingenuity, I found the way to shape an ancient vase, placing the foot in the acute de vono, and the neck, and the mouth, with the machines on top of the roundest part of it. The manner before shapes the man thus shall be. Let a cross of two lines be made, and the flat line shall be parted in equal parts, and the lead line shall be of parts ix, and leave four parts above, and five below, and in the middle shall be the centre A, and take axes in the four parts, making a half circle: the sides of which shall be C, and afterwards a point of the compass shall be placed at the end of the line B, and the other piotta at the opposite part C, circling downward: And thus let the right and left part be made, so that the acute angle below shall take the five parts: then two plumb lines shall fall at the fourth part of the diameter, where they shall intersect the curved line at the lowest part, and be stung. Then placed one point of the sixth at the point O, and the other point at one of them of the curved line, and circling down and back up to the other, the vouo shall be formed: and of the part which shall remain below, it shall be by the foot. And the neck and the mouth shall take two parts, and two the half circle: and thus shall the nine parts of the line be dispensed, and the manikins and the lid shall be at the will of the homo espero.
In another way it will be possible to make a vase by making a cross in the same way, and the flat line of parts say to us, and the plumb line of parts eight, and placing the compass with one point at the B, and the other point at the C, taking seven parts, and circling low so from one side as from the other will be the curved lines at the end of the plumb line at the bottom: Then, when two lines fall at the two inner parts A, to the bottom, where they touch the curved lines, there shall be the term to form the tip of the vessel, one of the sixth at the E, and the other at the said term circuiting to the other side: to form the merge of the vessel under which the foot shall be. Then, putting a point of the compass to the point A, and circling upward, up to its line, so that, from one side as well as from the other, the body of the vase is formed, and the throat with the touch will occupy two parts, then making them manicles, and other ornaments well placated.
Other different noses: of those of the past they may be made, but to form the present hereunder will make the same cross, but of parts twelve will be the flat line, and the lead line will be of parts eight, and first at the two parts more near the cross two lead lines of the same length as the middle one will fall, and at the two parts more near the cross two lead lines of the same length as the middle one will fall, and the comet will be placed near it with one point at the B, and with the other at the i, and circling downwards will be the hysterym of the middle line: and so from the other B to point z. Then, at the point between t and A, the tip of the sixth and the other tip at point i shall be the fourth part of a circle, and at the other point A, the second part of a circle. which will occupy one part, and two for the neck, and miss them, then coming down, a point will be put from the systems above point C, widening the compass two parts, on the curved line at point 3. and circling up to 4 will be the bottom of the nose, under which you will then be the foot, as seen below.
Another way of noses plus hands can be extracted from the circular shape by making a cross partia in parts six. First you will be the perfect circle, and the half circle will be for the bottom of the vase by adding a part more, yes for the bottom of the vase by adding a part more, yes for the bottom of the vase by adding a part more, yes for raising it to the quantum, as if to have a field to decorate it: another part you will be at the neck, and with another part you will be at the neck, and with another part at the lid by tightening those parts that are drawn below, and your foot will be as high as a part beyond six. And though I have given you a rule, and a way to make you strong of vases, yet with the same rules you could well make them all different, and at most beautiful ornaments, of which you could dress, which I did not wish to do, so as not to hinder the lines.
In different ways they can make oval shapes, but in four ways I'll make the rule. For this shape shown here, the sooner the shape will be shown here, the sooner you will have two perfect triangles of equal sides joined together, on the sides of which will be drawn four lines which will be 1. 2. 3. 4., and the centers to make this shape will be four A, B, C, D, starting from which center you want, but you will put a tip of the sixth at point B, and the other at point one, and pulling the circle up to 2. Then the point A is admitted a point, and from point 3 to 4 the sixths are drawn, then at point D one point is placed and the other 2, a, 4 pulling the circle, and so at point C, the same point and from 1 to 3 pulling the circle will be formed the oval shape. And when this longer shape will be made, the same circular lines with the same points should be drawn, always keeping at the bottom. And when we will make this form more round are drawn the circular lines as far away from the centers as it will have to be its size, and will always be the form more suited to the round, but it will never be perfect circle, to have more than one center.
For this second figure we will first make three circles in the way shown below, pulling the four straight lines, its centers will be I, K, L, M, and placing one point of the sixths at K, and widening the other point up to 1. Then pulling the circle up to 2 and so at point I one point of the sixths will be placed, and the other point at 3 pulling the circle up to 4 will be formed the oval shape and this shape resembles the natural egg.
For the third oval form shown below the way to do it will be, that two perfect squares are made joined together, and pulled the lines in the middle of them will be two centers G, H, and the other two centers will be E, F, either put the tip of the sixth at the F, and the other tip the 1 by pulling the circle up to the 2. Then the same is done at the center E, and from the 3 to the 4 is the circuit below placed the compass at the center G, and enlarged up to the 1. Turning up to the 3, and the same from the center G and enlarging the compass up to the 2 and pulling up to the 4 will be done the form here below drawn.
If you want to form this fourth oval figure, there will be two circles, one touching the center of the other at the corners of the curved lines will be two centers N, O, and at the centers of the circles will be two other centers I, P, Q, and pull the continuous lines from center to center, one point of the compass in the center O and the other point at 1 pulling the curved line up to 2 then one point of the sixth in the center N, the other point from 3 turning up to 4, and will be formed this oval figure, which is very grateful to the eye and to be used for the ease of doing it and for its sweetness.
After the circular form, there are many forms that tend towards the octagon, that is, the octagon of eight faces, the hexagon, that is, six faces, the pentagon of five faces, and below we can make different forms of several sides, which all tend towards roundness: but at present we will deal with this three main ones, which are more on the subject.
This octagonal form will get out of the perfect picture by first pulling the two lines at the junction, and placing one point of the sixth at one corner of the square, and the other point at the center of it, and turning d at the two sides of it square, as if it were the fourth part of the circle; and doing so at the four corners where the curved lines intersect with the sides of the picture, there will be the real terms of the octagon form. And though from the circle this anchor could cut out by making a cross, and each fourth part divide by half, which will be eight parts, that would be at the fourth beggar's fourth, but this would be brought by art.
The shape is hexagonal, i.e. six faces, it will be like this. A circle is made without widening or tightening the compass, but above the circular compassing line, where the points will touch, there rightly will be six points, so that from this point to the other drawn a line, the six faces will be formed. And this is the origin of the name of the compass, which in many places in Italy is called the sixth, to be the semidiameter in the sixth part of the circumferentia.
To form this pentagonal figure, i.e. of five sides, it is not as easy as the other, to be of odd sides, and of more numbers than three, nevertheless it will be so theoretically. Once a perfect circle has been made, inside that one there will be a cross, that is a flat line which is the diameter, and above the diameter a plumb line will fall. Then from the left side the half diameter will be divided into two equal parts, which will be 3, and from the one at the top, which will be a cross, the compass will be widened, and from the cross down to above the diameter it will be a circuit, not moving the tip of the sixths from 3. and where the curved line will fall from the cross above the diameter, what will be from 2 to the cross, that will rightly be one of the five sides of the pentagon. In that figure there are still the ten faces, so that from the center to the number 2 will be one side of the ten faces, and more this figure bears, still the side of the sixteen faces, but from the circumferentia at 1 towards the center at point 2 will be one of the sides of the sixteen faces.