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 can∣not be divided into parts, because it containeth neither length nor bredth in it.

A Line.

A Line is a right consecutive Imagination in length, be∣ginning 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 clo∣sed, 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 Per∣pendicular, 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 Tri∣angle, 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 Qua∣drangle 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) Qua∣drangles.

WHen four even long streight lines are joyned toge∣ther 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 afore∣said 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 af∣ter 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 Com∣pass 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 Com∣pass, 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 Cen∣ter 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 di∣videth 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 Quadra∣tus: then if you draw a streight line from the one cor∣ner to the other, it is called Diagonus: because it divi∣deth the four corners into two even parts.

[illustration]

NOw when a Workman hath seen a forme of some of the most necessary Superficies, he must proceed fur∣ther, 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 afore∣said, 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 o∣ther, 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 touch∣eth 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 a∣bout 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 de∣clared, 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 e∣ven 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, cor∣ner wise: and then the Quadrangle B. C. D. E contain∣eth 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. con∣tain 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 great∣ness 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 Tri∣angles 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 tri∣angles A. G. K. and G. C. E. are also of one greatness: so that those that are cut off, and those that are added there∣unto, are of one quantity. By these alterations afore∣said a man may easily measure how many feet, ells or roods foursquare, are contained in a three-corner'd Su∣perficies.

BUT it falleth out, that a Triangle, (which is three cor∣ner'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 desi∣reth any proof hereof, may take each piece, and alter it into a Quadrangle, and after into a Quadrate, as hereaf∣ter shall be shewed, and he shall find it true.

[illustration]

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 draw∣ing 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 cor∣ner'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 fol∣lowing. First then, seek the greatest Quadrate or Qua∣drangle 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 fol∣lowing may be seen.

LEt by Example your many corner'd Figures first be marked with the great Quadrangle with these Let∣ters 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 a∣foresaid Quadrangles. Touching the Triangles, when you have changed the same (according to your former instru∣ction) 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 Qua∣drangle O. P. Q. R. set upon it, and the Diagonal line is placed behind the greater (which is L. M. T. S both mark∣ed 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.

[illustration]

FOr Example of the figure aforesaid, I suppose, Hou∣ses 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 Cor∣nice, 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 al∣though the Column should be a Dorica (yet it is to be un∣derstood 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 suf∣fice 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 a∣gainst 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 spa∣ces 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 four∣ty, and the space contains ten times ten, which is an hun∣dred. The o••her line upon the two longest sides, con∣tains 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.

[illustration]

IF a man should make three Points, (which should not stand upon a right line) and desiring to have a circum∣ference 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 super∣ficie 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, ma∣ny Pillars or Columns, which beneath in the joynts at the-Bases are broken asunder, which is, because their Ba∣ses 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 o∣ther, 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 high∣est 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 to∣ward 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 inner∣most 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 with∣in the small Circles: the other members and Ornaments are alwayes to bee made, according to the will of the inge∣nious workman.

[illustration]

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 neer∣er rule, to make the forme of an Egge, then Albertus Du∣rens 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 Per∣pendicular 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 ac∣cording 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 touch∣ing the sides 3. and 4. then the bottom of the vessel will be closed up: then place the Compass upon the point be∣tween I. and A. and it will be the roundness of the vessel above: the other four parts serve for the neck of the ves∣sel, with the rest of the work.

A Man may make a vessel only by a Circular forme, making therein a circular cross, and dividing eve∣ry 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 ac∣cording to the rule aforesaid, a man may make an infinite number of vessels, and a man may alter them by their Or∣naments, 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 a∣gain, 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 fi∣gure 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 plea∣sure 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.

[illustration]

TOuching the Circles there are many Figures which are round, and yet some have 5. 6. 7. 8, 9. and 10. cor∣ners, &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 di∣viding 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 easi∣est made in a Circle: for when the Circle is made, you may divide the Circumference in six parts equally, with∣out 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 afore∣said 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 al∣so make a sixteen-corner'd Figure out of this wideness 3. 4 and place a particular line upon the point 1. And Al∣bertus 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 Circum∣ference into unequal parts, how many soever they be: but not to bring the Reader into confusedness, with ma∣king of many formes, I will only set down this divided in∣to 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 al∣so 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 re∣quired.

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 Qua∣drate.
• THe second forme or figure in Latin, is called Sexquiquar∣ta, 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 Com∣pass and so drawn, will shew you the length of the Dia∣gonal 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 un∣to 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 joyn∣ed 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 Work∣mans hand, which is, that a man should lay a sieling of a house in a place which is fifteen foot long, and as ma∣ny 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 cor∣ner 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-cor∣ner'd pieces marked E. B. G, and C. F. and C. left for some other use.

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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 Geome∣try, 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 low∣ermost 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 Perpendicu∣lar lines, which Perpendicular lines shall shew the wide∣ness 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 al∣though 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 di∣viding is innumerable, yet for this time. lest I should be too tedious, I here end my Geometry.

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FINIS.