An Excellent introduction to architecture being a book of geometrical practice, which is the first degree of all arts : wherein is contained variety of examples of that admirable science ...

A Point.

FIrſt, you muſt underſtand that a Point is a Prick made with a Pen or Compaſs, which cannot be divided into parts, becauſe it containeth neither length nor bredth in it.

A Line.

A Line is a right conſecutive Imagination in length, beginning at a Point, and hath no bredth.

A Parallel.

WHen two lines are ſet or placed a little diſtance one from the other, thoſe two Lines, according to the Latin phraſe, are called Parallel, and by ſome Equidiſtances.

Superficies.

WHen theſe two Lines aforeſaid are encloſed at each end with other Lines, it is then called a Superficies, and in like ſort all ſpaces, in what manner ſoever they are cloſed, are called Superficies or Plains.

Perpendicular.

WHen there is a ſtraight upright Line placed in the middle of a croſs ſtreight line, then it is called a Perpendicular, or Catheta line, and the end of the Croſſes or ſtreight line on both ſides of the Perpendicular are called ſtreight Corners.

Acutus, Obtuſus.

WHen a leaning or ſtreight Line is placed upon a ſtreight line, without Compaſs or Equallity, as much as the ſame line bendeth, ſo much ſhall the corner of the ſtreight line be narrower below, and the other ſo much broader as a right and even corner, the ſtraight corner in Latin is called Acutus, which ſignifieth ſharp, and the wider Obtuſus, which ſignifieth dull.

Pyramidal.

A Corner or Point called Pyramidal, and alſo Acutus in Latin is when two even long ſtreight lines meet or joyn together at the upper end, as the Figure declareth.

Triangle.

When ſuch a Figure, as aforeſaid, is cloſed together at the foot with a long ſtreight line; it is then called a Triangle, becauſe it hath three ſharp corners.

2. Triangle.

WHen a Triangle with two even ſtreight lines, is cloſed together with a longer line then theſe two are, it ſhall have ſuch forme as you may ſee in the Figure of the third Triangle.

3. Triangle.

A Triangle which is made of three unlike lines, will alſo have three unlike corners.

Quadrangle.

WHen two long and two direct down-right lines are joyned together at the four corners, it is called Quadrangle with even ſides or corners, but when the four lines are all of unlike and contrary length, then it is a Quadrangle of uneven ſides, as the Figure ſheweth.

YOu muſt note, that although all four corner'd Figures may be called Quadrangles; nevertheleſs, for that the direct four corner'd Figures are called Quadratus, for difference from them, I will name all Figures which are like unto a Table (that is, longer then broad) Quadrangles.

WHen four even long ſtreight lines are joyned together at the corners, they are called Quadratus; which are four corner'd: when you make the two corners therof ſharp; and the other two corners ſomewhat blunter, then it is called a Rombus.

ALthough you may turn and make all the Figures aforeſaid right four ſquare: yet the Workman may find other Figures with divers corners, the which (as I will hereafter ſhew) he may make four ſquare.

WHen a man with his compaſs draweth a bowe, and after that he draweth another bow right againſt it, that is called a Superficies of crooked lines, with two like corners: and then draweth a ſtreight line from the one corner to the other: and from one point or center where the Compaſs ſtood to the other, another ſtreight line; thereby you ſhall find the right four parts thereof.

BUt if a man draw a whole right line with his Compaſs, that is called a full Circle or round Superficities, and the point in the middle is calld the Center, the utmoſt line is called Circumference: and if you draw a ſtreight line through the Center, it is called a Diameter: becauſe it divideth the Circle in two even parts.

WHen the half Circumference is cut through the Center of the Diameter, then it is called half a Circle: and if you make a ſtreight line upright in the half Circle, then that line maketh two even quarters of a Circle, and divideth the Diameter alſo into two half Diameters.

WHen a man draweth four even long lines, and joyneth them together, they make a perfect corner'd Quadratus: then if you draw a ſtreight line from the one corner to the other, it is called Diagonus: becauſe it divideth the four corners into two even parts.

NOw when a Workman hath ſeen a forme of ſome of the moſt neceſſary Superficies, he muſt proceed further, and learn to augment or diminiſh the ſame, and to turn them into other formes: but yet in ſuch ſort, that they may have even parts in them.

AND firſt, if out of the length of the Diagonus aforeſaid, by the adding of three other even long lines, he maketh another four ſquare: that four ſquare ſhall be once as great again as the firſt, which is to be underſtood in this ſort: That the four ſquare of A. B. C. D by the Diagonus is divided into two Triangles, and the greater four ſquare A. D. F. E. containeth four ſuch Triangles: but for that the two firſt four ſquares hang one within the other, therefore for the better ſhewing thereof, they are here once again ſet down ſeverally: whereby you may ſee that the Quadrate G. (as I ſaid before) containeth two Triangles, and the Quadrate H. containeth four ſuch Triangles, ſo that the proof thereof is clearly to be ſeen.

IF within a four ſquare you make a Circle which toucheth the four ſides of the ſaid four ſquare, and without the ſaid four ſquare another Circle which toucheth the corners marked A. B. C. D. Then the outmoſt Circle muſt be once as great again as the innermoſt: and then if about the greateſt Circle you make another four ſquare as C. D. E. F. then the two four ſquares muſt in like ſort be once as great again as the other. The proof whereof ſtandeth hereby marked with the Letters K. L. for clearer underſtanding of the ſame.

BY this alſo the projecture or the foot of the Baſes of the Tuſcane Columns or Pillars, and alſo the bredth of the foundation of them underneath by Vitruvius declared, is ſet forth.

THe Workman muſt yet proceed further, and learn to know how to change a Triangle into a Quadrangle, and alſo at laſt bring it to a right Quadrate, to the which I will ſet down divers formes. Firſt, take a Triangle with even corners, as A. B. C. and divide the Baſe, (which is the name of all lower lines) B. C. in two even parts, and there place the Letter E. Then from the point E. to A. draw a line, which will divide the Triangle into two even parts. Then if you take that part which is marked A. E. C. and joyn it to the other part, marked A. E. B. it will make a Quadrangle, as A. D. B. E, made of a Triangle.

YOu may alſo change this Triangle in other manner, dividing the lines A. B. and A. C. each in two like parts as F. and G. Then draw a line through D. E. as long as the Baſe B. C. Then ſhut up the two Equidiſtances, corner wiſe: and then the Quadrangle B. C. D. E containeth ſo much in it as the Triangle A. B. C. and the proof thereof is, that the two Triangles B. C. F. and G. E. C. contain ſo much in them, as the two other Triangles A. F. H. and A. I. G.

A Triangle with even points, may be divided thrice into two equal parts, dividing each ſide in two parts, as in the Figure P. Q. R. it is ſeen through the three lines, which on either ſide make two great Triangles.

THe ſame Triangle P. Q. R. may thus be changed into a Quadrangle: divide the ſide P. Q. and the ſide P. R. each in two equal parts, then draw a line C. T. as long as Q. and R, and then draw a line direct downward from T. R. to cloſe it up: and then that Quadrangle contains as much ſpace within it as the Triangle aforeſaid, becauſe that the Triangle which is cut off P. S. V. is of like greatneſs with the other Triangle marked V. R. T.

ANd although there is a Triangle of unequal ſides, yet a man may make it a Quadrangle, in ſuch ſort as I ſaid before of the right Triangle: for although the two Triangles that are cut off, and thoſe two that are added unto it, are not of one greatneſs, yet the Triangles A. F. L. and B. D. F are one as great as the other: and again, the triangles A. G. K. and G. C. E. are alſo of one greatneſs: ſo that thoſe that are cut off, and thoſe that are added thereunto, are of one quantity. By theſe alterations aforeſaid a man may eaſily meaſure how many feet, ells or roods fourſquare, are contained in a three-corner'd Superficies.

BUT it falleth out, that a Triangle, (which is three corner'd) ſuperficie or plain, muſt be parted croſs-wiſe in two equal parts: then out of one of the ſides that you will cut through, you muſt make a right four ſquare, as from the ſide A. B. and draw therein two Diagonus from corner to corner, which will ſhew you the Center C. and draw one Circle through that three-corner'd part, which you will divide, and ſo you ſhall finde the two points, where you ſhall draw your dividing line. He that deſireth any proof hereof, may take each piece, and alter it into a Quadrangle, and after into a Quadrate, as hereafter ſhall be ſhewed, and he ſhall find it true.

AN Architector muſt alſo undergo other Burthens, for that he muſt know how to divide a piece of ground, that no man may be hindred thereby. As for Example, if there were a piece of ground that lay three-corner'd wiſe, with unequal parts, having on the one ſide thereof a Well, but not in the middle: and this ground, or three-corner'd piece of Land is to be divided into two equal parts, in ſuch ſort, that each of them may have the uſe of the Well: it muſt be done in this manner. I make a Triangle marked A. B. C. and the Well is marked with G. Now divide the line B. C. with a dark line in the two equal parts as the Letter D. ſheweth, and then drawing a line from D. to A. then the Triangle is divided into two equal parts: but both of them cannot yet come to the Well: then draw another line from the Well G. to A. and from the point D. you muſt ſet an Equidiſtancie againſt G. A. marked with E. and drawing from G. which is the Well: the black line to the letter E it will divide the ground in two even ſeveral parts, and each of them ſhall have the VVell at the end of his Ground, for that part A. B. G. E. containeth in it juſt as many feet or roods, as that part which is marked G. E. C.

I Shewed before, how a man ſhould make a four ſquare Superficies once as great again as it is, but it may fall out, that a man is to make it but half as great again, or more or leſs, as he thinketh good, or as occaſion ſerveth, which the Architector is alſo to learn of neceſſitie. Which to ſhew, I ſet down a right four ſquare thing marked A. B C, D, which I will have three quarters greater: the ſame three quarters I ſet by the ſide thereof, ſo that the ſame with the Quadrate together make a Quadrangle A. E. C. G. To bring this Quadrangle into a right Quadrate, you muſt lengthen the line A. E. yet a quarter longer, or from the ſide of the Quadrangle E. G, and place F. there: then upon the line A. F. make half a Circle: which line will ſhew you the one ſide of the Quadrate which you ſeek for: which Quadrate being made, will contain as much in it as the Quadrangle already made And in this manner you may change all Quadrangles which are long four corner'd pieces of work, into a juſt and true Quadrate.

NOw to prove that which I ſaid before, you muſt join the Quadrangle with the Quadrate together, in one ſquare Superficie, as Q. R. S. T. and from the corner R. to the corner S. draw a Diagonus, and it is certain that that Diagonus will make two even parts. Now Euclides ſaith, that when a man taketh any even parts from even parts, the reſt of the parts alſo remain alike: then take the Triangle K. L., and the Triangle M. N. which are both alike: the right four corner'd ſuperficie P. is of the ſame greatneſs, that the longer ſuperficie O is.

AGain, you may eaſily change a Quadrate into a Quadrangle, as long or as narrow as you deſire to have it, doing thus; Make your Quadrate A. B. C. D and lengthen your Line A. B. and the Line B. C. Which done, then ſet the length of the Quadrangle, which you deſire to have upon the line A. G. Then from the point G. draw a line alone by the corner of the Quadrate D. to the line C. F. and there you find the ſhorteſt line of the Quadrangle: and ſo to the contrary you ſhall by the leaſt ſide of the Quadrangle finde the longeſt alſo, as you may alſo prove by the aforeſaid Figure: for when you take away the Triangles M. N. and O. P. which are both alike: then the two parts which are K. L. are alſo alike.

AN Architector may by chance have a piece of work of divers unequal ſides come to his hands, which he is to put ino a Quadrangular or Quadrate forme, to know what it containeth, and ſpecially when it belongeth to more then one man, whether it be Land or any other thing. For although the Architector or Surveyor of Land could not skill of Arithmetick or Ciphering: yet this rule cannot fail him, nor any other man that deſireth to find out the deceit of a Taylor. Thus, I ſay then, let it be what forme ſoever it will, I ſet down this hereafter following. Firſt then, ſeek the greateſt Quadrate or Quadrangle that you can take out of it: that done, ſeek yet another Quadrate or Quadrangle, as big as you can take out of it, out of the reſt of the ſaid work: and if you can after that make more Quadrates or Quadrangles out of it, I mean all with right corners, take them out alſo: but if you can find no more in it, then make Triangles alſo as big as •• u can, of which Triangles (as you are taught before) you may make Quadrangles, and let every piece ſeverally be marked with Characters, as in the Figure following may be ſeen.

LEt by Example your many corner'd Figures firſt be marked with the great Quadrangle with theſe Letters A. B. C. D. and then with a leſs Quadrangle, as E. F. G. H. the reſt are all Triangles. Now ſet the greateſt Quadrangle L. in a place by it ſelf, and then the other marked with M. which ſet upon it, that the two corners or ſides may be alike: which done, lengthen the line E. F. and the line E. G. and where they ſtay or touch under the great Quadrangle L. there ſet an I, from this I. a Diagonal line, being drawn through the corners B. H. the ſame line ſhall be drawn to the point: that, by the ſhutting of the Characters B. M. L. D. will ſhew you another Quadrangle, of the like quantity that the Quadrangle M. is: ſo that the whole Quadrangle D. C. L. M. containeth the two aforeſaid Quadrangles. Touching the Triangles, when you have changed the ſame (according to your former inſtruction) into Quadrangles, as you may ſee by the Triangle N, ſo may you put that Quadrangle alſo in the greateſt Quadrangles (for leſs trouble.) The great Quadrangle A. L. M. C. is once again placed above with the ſmall Quadrangle O. P. Q. R. ſet upon it, and the Diagonal line is placed behind the greater (which is L. M. T. S both marked with N. ſo that the Quadrangle A. C. S. T. containeth three Quadrangles L. M. N. and as many more as there are: you may in this ſort bring them all in one Quadrangle: if there falleth out any crooked lines, the skilful Architector or Workman may almoſt bring them into a ſquare, and thoſe Quadrangles, if need be, may alſo be reduced into perfect four ſquares, as aforeſaid.

WHen a man hath a line or other things of unequal parts, and there is alſo another longer line, or ſome other thing, which a man would alſo divide into unequal parts, according to the proportion of the ſhorter line, then let the ſhorteſt line be A. B. and the greateſt line A. C. now it is neceſſary that from the uppermoſt point A. you ſhould make a corner, as A. B. and A. A. Then take your longer line, and ſet it with the end C. upon B. and let the other end reſt at the hanging line A. A. then from every point of the uppermoſt line A. B. let a hanging line fall upon the line A. C. ſo that they may be equidiſtant with the line A. A. and where the ſaid lines cut through each other, there is the right diviſion proportioned according to the ſmaller. This rule ſhall not only ſerve the Architector for many things, as I will partly ſhew: but will alſo ſerve many Artificers to reduce their ſmall works into greater.

FOr Example of the figure aforeſaid, I ſuppoſe, Houſes or Pieces of Land to be of divers wideneſs, which ſhould be narrower before then behinde. Which Houſes by Fire or War are ſo decayed, that in the forepart between C. D. there were but ſome ſigns of diviſion to be ſeen of the houſes, and behind the houſes between A and B. no ſign at all to be ſeen. Now as the misfortune was paſt, and that every man deſired to have his part of his inheritance, then the Architector, as an Umpire, according to the rule aforeſaid, ſhould divide the longeſt line according to the proportion of the ſhorteſt, to give every man his own: as you may ſee by this Figure following.

THE Architector muſt have a well-proportioned Cornice, which if he would make greater, keeping the ſame proportion, he may do it as he is formerly taught, as in this Figure following is ſhewed by the ſhort line marked A. B. and the longeſt line marked A. C.

AN Architector or Workman muſt likewiſe learn to augment and make greater a hollowed Column, which he may alſo do by the two lines aforeſaid, and although the Column ſhould be a Dorica (yet it is to be underſtood of all kinds of Columns. This rule will alſo ſerve (not only for the three Figures ſet down) but alſo for as many, as if I ſhould ſhew them, it would contain a whole book of them alone, and therefore this ſhall ſuffice at this time for the Workman.

THe further that any material thing ſtandeth from our ſight, ſo much it ſeemeth to leſſen, and diminiſh by means of the Air, which conſumeth our ſight: therefore when a man will make or place one thing above another, againſt any place or wall, and would have the ſame thing to ſhew above in the middle, and beneath, as great in one part as in the other, it is convenient for him to follow this rule, which is, for that our ſight runneth in circumference: therefore a man muſt firſt chuſe the place, from whence he will ſee the ſame: there placing a Center, and then draw a quarter of a Circle from your eye upwards. Which dividing in even parts, you ſhall, by the lines that go out of the Center through the Circle againſt the wall, finde the unequal parts: the which although upwards againſt the wall, they ſhall ſeem greater: yet in your ſight they will ſhew all of one greatneſs. By this rule you may alſo meaſure heights, aiding your ſelf with the numbers.

MAny men are of opinion, that ſtreight lines, in what manner ſoever they are cloſed, contain as many ſpaces one way as another, (that is to ſay) if a man had a cord of fourty foot long, and ſhould lay it diverſly, in a round, long, three corner'd, four ſquare, or five-corner'd forme: but the ſuperficies are not of one ſelf-ſame ſpace, which may be ſeen by theſe four ſquare Figures following; for the firſt line holdeth on either ſide ten, which is fourty, and the ſpace contains ten times ten, which is an hundred. The o • her line upon the two longeſt ſides, contains fifteen ſpaces, and on the ſhorteſt ſides five, making fourty alſo: but five times fifteen make but ſeventy and five.

IF the Quadrate ſtretcheth further out, ſo that the two longer ſides were eighteen a piece, then the ſhorteſt ſides muſt each have two to have fourty upon the line, but the ſpace ſhould contain but ſix and thirty. And hereby you ſee what a perfect forme may do againſt an imperfect. And this rule the Workman ſhall uſe, that be may not be deceived, when he will change one forme into another.

IF a man ſhould make three Points, (which ſhould not ſtand upon a right line) and deſiring to have a circumference made, the Compaſs muſt paſs along upon each of theſe Points. To do it from the point one to the point two, he muſt draw a line, and from the point two to the point three another: which two lines ſhall each of them be divided into two equal parts, and ſetting the ſquires half way in them, as you ſee it in the Figure, by that Croſs it will ſhew you the Center, wherein you muſt ſet one foot of the Compaſs, and with the other draw the Circle through all the ſaid points.

YOu may find the Center of three points another way, without your Compaſs, making a two-corner'd ſuperficie from the one point to the other, through the which corners two ſtraight lines being drawn long enough downwards where they croſs one over the other, they will ſhew you the Center of the three points.

BUt for that a Workman holds this to be a ſuperfluous ſpeech, and a thing of no moment, it may be that a Workman may have a piece of a round work to do, which he is to perfect and make full round, by this rule he may find the Center, Circumference, and Diameter thereof, as the Figure ſheweth.

WE finde in Antiquities, and alſo in modern works, many Pillars or Columns, which beneath in the joynts at the-Baſes are broken aſunder, which is, becauſe their Baſes were not well made according to their corners: or elſe, becauſe they are not rightly placed: ſo that they have more weights upon them on the one ſide then on the other, whereby the Cantons break, which the Workman, by knowledge of the lines, and help of Geometry, may prevent in this manner: that is, he muſt make the pillar round underneath, and his Baſe hollow inward: ſo that when you place the Pillar by the lead, it may preſently ſettle it ſelf without any hurt. To finde this roundneſs, you muſt ſet the one point of the Compaſs upon the higheſt part of the pillar that is under the A. and the other point thereof upon B. and then draw or winde it about to C. and that ſhall be the roundneſs, making the hollowing of the Baſe, according to the ſame meaſure: you may do the like with the Capital, as you may ſee in the Pillar by it.

IF a Workman will make a Bridge, Bowe, or any other round Arched piece of work, which is wider then a half Circle, although Maſons practiſe this with their lines, whereby they make ſuch kinde of works, which ſhew well to mens ſight, yet if the Workman will follow the right Theorick and reaſon thereof, he muſt obſerve the order heretofore ſhewed. When he hath the wideneſs of the height, then he muſt make half a Circle out of the middle: after that, upon the ſame Center, he muſt make another leſſer Circle, which muſt be no greater then he will make the height of the Bowe, or Arch: then he muſt divide the greateſt Circle in equal parts, which muſt all be drawn with lines to the Center: then you muſt hang out other Perpendiculars upon your Lead: and where the lines that go to the Center cut through the leſſer Circle, from thence you muſt draw the croſs lines toward the Perpendicular, and where they cloſe together, there the Bowe or Arch which is made, ſhall be cloſed: as by the points or pricks here under is ſhewed.

BUt if you deſire to make the Bowe or Arch lower, then you muſt follow the rule aforeſaid, and make the innermoſt Circle ſo much leſs, which is to be underſtood, that the more parts that you make of the greater Circle, ſo much the eaſier you ſhall draw the crooked lines which you would have: from this rule there are many others obſerved, as hereafter you ſhall ſee.

CAlling the former rule to minde, I deviſed the manner how to forme and faſhion divers kinds of veſſels by the ſame, and I think it not amiſs to ſet down ſome of them: This only is to be marked, that as wide as you will make the veſſels within, ſo great you muſt make the inner moſt Circle. The reſt, the skilful Workman may mark by the Figures, that is, how the lines are drawn to the Center, and the Parables, and out of the ſmall Circle. The Perpendiculars hanging, the veſſels are formed: the foot and the neck may be made as the Workman will.

BUt if you will make the body of the veſſel thicker, then you muſt make the half Circle ſo much the greater, and make the belly hanging down under it, to touch the great Circle, by the falling of the Perpendiculars upon the croſs line, as by theſe Figures 3.4.5. it is ſhewed: whereby a man by this meanes may make divers veſſels, differing from mine. The necks and covers of theſe veſſels are within the ſmall Circles: the other members and Ornaments are alwayes to bee made, according to the will of the ingenious workman.

IT is an excellent thing for a man to ſtudy or practiſe to do any thing with the Compaſſe, whereby in time men may find out that which they never imagined: as this night it happened unto me, for that ſeeking to find a neerer rule, to make the forme of an Egge, then Albertus Durens hath ſet downe, I found this way to make an Antick veſſell, placing the foot beneath at the foot of an Egge, and the necke with the handles above upon the thickeſt part of the Egge. But firſt you muſt frame the Egge in this manner: Make a ſtreight croſs of two lines, and divide your croſs line in ten equal parts: that is, on each ſide five. Then, ſet the Compaſs upon the Center A, and with the other foot thereof, draw in two parts, that is, to C. making half a Circle upwards. That done, ſet one foot of the Compaſs upon the point marked B. and with the other draw in the uttermoſt point C. drawing a piece of a Circle downwards toward the Perpendicular, and doing the like on the other ſide, you muſt make a point below. Then take the half of the half Circle above that two parts, and place it at the undermoſt point of the Perpendicular upwards above O, where the Center to cloſe the Egge, ſhall ſtand: the reſt under ſhall be for the foot: the neck, without doubt, may be made two parts high, and the reſt according to the Workmans pleaſure, or according to the Figure here ſet down.

YOu may alſo make another forme of a Cup or veſſel, after the rule aforeſaid. But from the point A. (which doth ſhew the bredth of the foot, and the wideneſs of the mouth) you muſt make your Circle upwards from C. unto the two Perpendiculars, where the body ſhall be cloſed up. The neck ſtanding above it ſhall be two parts high: but the reſt of the Workmanſhip ſhall be made according to the will and device of the Workman.

BY this means you may make other different kinds of Cups or veſſels: but theſe that follow, you muſt make in this ſo • t: you muſt divide your croſs line in twelve parts through the point A. making two Perpendiculars to ſhew the foot and the neck: then ſetting one foot of the Compaſs upon B. and the other foot upon I. drawing a piece of a Circ • e downwards towards the Perpendicular: and the like being done on the other ſide to the Figure of 2. then place your Compaſs upon the point C. and touching the ſides 3. and 4. then the bottom of the veſſel will be cloſed up: then place the Compaſs upon the point between I. and A. and it will be the roundneſs of the veſſel above: the other four parts ſerve for the neck of the veſſel, with the reſt of the work.

A Man may make a veſſel only by a Circular forme, making therein a circular croſs, and dividing every line into ſix parts: the half-circle ſhall be the belly of the veſſel, and a ſixt part upward for a Freeſe, that there may be more place to beautifie it: another part ſhall be the • eg • t of the neck, and another part the corner: and for the foot, although it be but half a part high, it may well go a ſixth part without the round: and although I have ſet down but ſix manner of cups or veſſel, yet according to the rule aforeſaid, a man may make an infinite number of veſſels, and a man may alter them by their Ornaments, whereof I ſay nothing that you may ſee the line the better.

A Man may make Oval formes in divers faſhions, but I will only ſet down four. To make this firſt Figure, you muſt ſet two perfect Triangles one above the other, like a Rombus, and at the joyning of them together, you muſt draw the lines through to 1. 2. 3. 4. and the corners A. B. C. D. ſhall be the four Centers, then ſet one foot of the Compaſs upon B. and the other upon I, and draw a line from thence to the Figure 2. After that, from the point A. and 3. to 4. you muſt alſo draw a line: which being done, ſet the one end of the Compaſs in the point C. and then draw a piece of a Circle from 1. to 3. and again, the Compaſs being in the Center D. draw a piece of a Circle from 2. to 4. and then the forme is made. You muſt alſo underſtand, that the nearer that the Figures come to their Centers, ſo much the longer they are: and to the contrary, the further that they are from their Centers, the rounder they are: yet they are no perfect Circles, becauſe they have more then one Center.

FOr the making of the ſecond Oval you muſt firſt make three Circles, as you ſee here drawing, where the four ſtreight lines ſtand: the four Centers ſhall be I. K. L. M. Then placing one point of the Compaſs in K. yon muſt draw a line with the other point from the Figure of 1. to 2. Again, without altering the Compaſs, you ſhall ſet the one foot of the Compaſs in I. and ſo draw a piece of a Circle from the figure 3. to the figure 4. and that maketh the Compaſs of the Circle. This Figure is very like the form of an Egge.

THE third forme is made by two foure corner'd ſquares, drawing Diagonen lines in them, which ſhall ſhew the two Centers G. H. and the other two corners E, and F. Then draw a piece of a Circle from F. to the figure 1. and ſo to 2. Do the like from E. to 3. and 4. which done from the points G. and H. make the two ſides from 1. to 3 . and from 2. to 4. and ſo ſhut up the Ovale.

IF you will make this fourth Oval, draw a line at pleaſure as A. B. then ſet one foot of the Compaſſes at C. and ſtrike a Circle, then remove the Compaſſes, and ſet one foot at D and ſtrike another Circle, then ſet one foot of the Compaſſes at E. and cloſe up the line from G. to H. then ſet one foot at F, and cloſe the line from • to K.

And although our Authour ſaith, there are four forms of Ovals; yet this laſt figure is of the ſame form as the firſt, only this is eaſier to make.

TOuching the Circles there are many Figures which are round, and yet ſome have 5. 6. 7. 8, 9. and 10. corners, &c. But at this time, I will ſpeak only of theſe three principally: becauſe they are moſt common.

THis Octogonus or eight points is drawn out of a right four corner'd ſquare, drawing the Diagonus which will ſhew you the Center: then ſet one foot of your Compaſs upon the corners of the Quadrate, and leading the other foot through the Center, directing your Circle toward the ſide of the Quadrate, there your eight points ſhall ſtand to make it eight corner'd, and although a man might only do it by the Circle, making a croſs therein, and dividing each quarter in two, yet it will not be ſo well, and therefore this is a ſurer and more perfect way.

THe Hexagonus, that is, the ſixt-corner'd Circle, is eaſieſt made in a Circle: for when the Circle is made, you may divide the Circumference in ſix parts equally, without ſtirring the Compaſs, and drawing the line from one Point to another, the ſix corners are made,

BUt the Pentagonus that is five-corner'd, is not ſo eaſily to be made as the others are, becauſe it is of an uneven number of corners, notwithſtanding you may make it in this manner: when the Circle is made, then make a ſtreight croſs therein: then divide the one half of the croſs line in two parts, as it is marked with the Figure 1. then place one foot of the Compaſſes at the Figure 1. and the other foot under the Figure 2. draw downward to the Figure 3. reſting that foot, and reaching the other to the aforeſaid place under 2. and you will have the length of every ſide of the Pentagonus. In this Figure alſo you ſhall find the Diagonus, that is, ten corners: for, from the Center to the Figure 3. that ſhall be one ſide thereof, you may alſo make a ſixteen-corner'd Figure out of this wideneſs 3. 4 and place a particular line upon the point 1. And Albertus Durens ſaith, that the ſame alſo will ſerve to make a ſeven-corner'd Figure.

THis figure will ſerve ſuch men as are to part a Circumference into unequal parts, how many ſoever they be: but not to bring the Reader into confuſedneſs, with making of many formes, I will only ſet down this divided into nine corners, which ſhall ſerve for an example of all the reſt, which is thus: Take the quarter of the Circle, and divide it into nine parts, and four of theſe parts will be the ninth part of the whole Circumference: you muſt alſo underſtand the ſame ſo, if you divide a Quadrate into eleven, twelve, or thirteen parts, &c. for that always four of theſe parts be the juſt wideneſs of your parts required.

THere are many Quadrangle Proportions, but I will here ſet down but ſeven of the principalleſt of them which ſhall beſt ſerve for the uſe of the Workman.

FIrſt, this forme is call'd a right four-corner'd Quadrate. THe ſecond forme or figure in Latin, is called Sexquiquarta, that is, which is made of a four-corner'd Quadrate, and an eighth part thereof joyned unto it. THe third Figure in Latin is called a Sexquitertia, that is, made of a four-ſquar'd Quadrate, and a third part thereof joyned unto it. THe fourth is called Diagonea of the line Diagonus: which line divideth the four-ſquar'd Quadrate croſs through the middle, which Diagonal line being toucht from under to the end thereof upwards with the Compaſs and ſo drawn, will ſhew you the length of the Diagonal Quadrangle: but from this proportion there can be no rule in number well ſet down. THE fifth Figure is called a Sexquialtera, that is, a four ſquare, and half of one of the four ſquares added unto it. THe ſixth is called Superbitienſtertias, that is, a four ſquare, and two third parts of one of the four ſquares added thereunto. THE ſeventh and laſt Figure is called Dupla, that is, double: for it is made of two four ſquare formes joyned together: and we finde not in any Antiquities, any forme that paſſeth the two four ſquares, unleſs it be in Galleries, Entries and other to walk in: and ſome Gates, Doors and Windows have ſtood in their heights: but ſuch as are wiſe will not paſs ſuch lengths in Chambers or Halls.

MAny Accidents like unto this, may fall into the Workmans hand, which is, that a man ſhould lay a ſieling of a houſe in a place which is fifteen foot long, and as many foot broad, and the rafters ſhould be but fourteen foot long, and no more wood to be had: then in ſuch caſe, the binding thereof muſt be made in ſuch ſort as you ſee it here ſet down, that the rafters may ſerve, and this will alſo be ſtrong enough.

IT may alſo fall out, that a man ſhould finde a Table of ten foot long, and three foot broad: with this Table a man would make a door of ſeven foot high, and four foot wide. Now to do it a man would ſawe the Table long-wiſe in two parts, and ſetting them one under another, and ſo they would be but ſix foot high, and it ſhould be ſeven: and again, if they would cut it three foot ſhorter, and ſo make it four foot broad, then the one ſide ſhall be too much pieced. Therefore he muſt do it in this ſort: Take the Table of ten foot long and three foot broad, and mark it with A. B. C. D. then ſawe it Diagonal-wiſe, that is, from the corner C. to B. with two equal parts, then draw the one piece there of three foot backwards towards the corner B. then the line A. F. ſhall be four foot broad, and ſo ſhall the line E. D. alſo hold four foot broad: by this means you ſhall have your door A. E. F. D. ſeven foot long, and four foot broad, and you ſhall yet have the three-corner'd pieces marked E. B. G, and C. F. and C. left for ſome other uſe.

IT hapneth many times that a Workman hath an eye or round Window to make in a Church, as in ancient times they uſed to make them, and he doubted of the greatneſs thereof, which if he will make after the rules of Geometry, he muſt firſt meaſure the bredth of the place where he will ſet it, and therein he muſt make a half Circle: which half Circle being incloſed in a Quadrangle, then he ſhall finde the Center by two Diagonal lines: then he muſt draw two lines more, which ſhall reach from the two lowermoſt corners above the Center, and touch the juſt half of the Circle above: and where the ſaid lines cut through the Diagonal lines, there you muſt make two Perpendicular lines, which Perpendicular lines ſhall ſhew the wideneſs of the deſired window: the liſt about it may be made the ſixth part of the Diameter, being round in bredth.

IF a Workman will make a gate or door in a Temple or a Church, which is to be proportioned according to the Place, then he muſt take the wideneſs within the Church, or elſe the bredth of the wall without: if the Church be ſmall, and have Pilaſters or Pillars within it: then he may take the wideneſs between them, and ſet the ſame bredth in a four ſquare, that is, as high as broad, in which four ſquare the Diagonal lines, and the two other croſs cutting lines will not only ſhew you the wideneſs of the door, but alſo the places and points of the ornaments of the ſame door, as you ſee here in this Figure. And although it ſhould fall out, that you have three doors to make in a Church, and to that end cut three holes, yet you may obſerve this proportion for the ſmalleſt of them. And although (gentle Reader) the croſs cutting thorow or dividing is innumerable, yet for this time. leſt I ſhould be too tedious, I here end my Geometry.

FINIS.