Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	line_n	perpendicular_a	3,095	5	14.0786	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A35744	Mr. De Sargues Universal way of dyaling, or, Plain and easie directions for placing the axeltree and marking the hours in sun-dyals, after the French, Italian, Babylonian, and Jewish manner together with the manner of drawing the lines of the signs, of finding out the height of the sun above the horizon, and the east-rising of the same, the elevation of the pole, and the position of the meridian ... / [edited] by Daniel King, Gent.; Maniére universelle pour poser l'essieu. English	Desargues, Gérard, 1591-1661.; King, Daniel, d. 1664?; Bosse, Abraham, 1602-1676.	1659 (1659)	Wing D1127; ESTC R17188	51,911	119

far as that which is lengthened POB one straight line DB even with the straight line DB of the II figure for it must reach unto it viz. In the Equinoctial at the point O and in an other place at an other time lengthen sufficiently III figure this straight line BD. Make in it the segments or cuttings even with the beams of the Sun of the I figure BD BF BC. Take in the III figure in the straight line POB conveniently a point I other than B. Make in the IIII figure three rods or sticks CI DI FI each of them sharp at both ends and equal with the three spaces CI DI FI of the third figure Draw a straight line along the Axeltree rod mark in this line of the Axeltree conveniently figure IIII one Cut BI equal with the space BI of the III figure Set figure IIII one of the ends of the stick CI to the point of shadow C one of the ends of the stick DI to the point of shadow D and one end● of the stick FI to the point of shadow F. Let the ends of those sticks or rods be so well fastened to the points of shadow CDF that they cannot stir Bring together the other ends I of those sticks in one point I. Put one of the point B of the Axeltree rod to the point of the pin B and the other point I with the three ends of the sticks CI DI FI set or joyned together And if you have been very exact in the work the point I of the pin will go and place it self with the three ends of the sticks set together in the point I if not you have not wrought exactly 6 To the Theoriciens It is no matter whether the Figures come right to the Compasses you are only to take notice what this insuing Discours ordains you to do MAke figure I with three straight lines CQRD DIPE and CF a Triangle even and like unto the Triangle figure III of the three points of shadow CDF upon the straight line CQRD figure I. make a Triangle CBD both like and equal with the Triangle figure III. of the Sun-beams CBD and upon the straight line FPID figure I. make a Triangle FDB like to the Triangle figure III. of the Sun-beams FDB make longer if need be figure I. on the side of D the straight lines CQRD and FPID By the points B and B draw a straight line BRAYH perpendicular to CQRD and a straight line BIAKL perpendicular to the straight line FPID find out the end or point A common to these two straight lines BRAYH BIAKL and by this end A draw a straight line AE perpendicular to the straight line BRAYH and a straight line AG perpendicular to the straight line BIAKL from the point R draw as far as the straight line AE a straight line RE even with RB from the point I draw as far as the straight line AG a straight line IG even with IB. From the point E carry to the straight line BRAIH a straight line EH perpendicular to the straight line RE from the point G carrry to the straight line BIAKL a straight line GL perpendicular to the straight line IG from the points B and B carry a straight line BQ that may divide in half the Angle CBD and a straight line BP that may divide in half the Angle DBF By the points Q and H draw a straight line QOH and by the points P and L draw a straight line POL find the end or the point O common to the two straight lines QOH and POL and from the point A for center and space AO draw an half circle that may meet with the straight lines AL in K and AH in Y. Now make in some other place even or flat as in the second figure in one and the same line BDFC three cuts BC BD BF even with the Sun beams figure III. BC BD BF each of them to his own from the point B of this second figure for center and from the interval or space EY or GK of the first figure draw an half Circle O from the point C figure II for center and from the space CO of the first figure draw an other half Circle O from the point D of the II figure for center and space DO of the first figure draw an other half Circle O and from the point F also of the II figure for center and space FO of the I figure draw an other half Circle O and if you have done right all these half Circles will meet in the same point O if not you have not been exact in working By the points B and O draw a straight line BO take in this line a point at discretion first make three rods even with the spaces CI DI FI of the second figure and every one sharp at both ends make in the length of the axeltree rod figure III the space BI even with the space BI of the II figure Lastly set these rods to the axeltree figure III as I have said at the end of the fifth table and the axeltree of the Dyal is placed There are some situations of superficies of Dyals where practising this manner of drawing one or the other of the points LH or O comes so far from the straight line CF that you should have need of too great a space to come to it But in what manner soever the superficies of the Dyal may be situated and at all times or seasons of the year I mean in any strange or odd kind of example that may be found you may work or practise these kinds of draughts with as much ease as in the most easy pattern 5 And by means of these three angles even with those in the air between the beams of the Sun you may chuse at pleasure within the lines that represent those beams other points CDF and otherwise disposed between them then those which the shadow of the point of the pin hath given upon the superficies of the Dyal and upon those three points chosen out at pleasure you may make an other triangle CDF and practise afterwards this manner of drawing as far as the triangle CBO figure II than in this triangle and in the straight line BC make BC BD BF even with the beams in the air BC BD BF of the third figure contained from the point of the pin B to the points of shadow CDF in the superficies of the Dyal each of them to his own and after you have taken as it is said the point I in the straight line BO you must make use of the points CDF last made in the triangle OCB for to set the rods CI DI FI to the axeltree BI then to work on as before To make other points instead of those of the superficies of the Dyal you need only to make some at the two extremities or furthest ends CF and make BC and BC equal one to the other and unequal with the middlemost BD but a little bigger

more or lesse according as the angles DBC DBF are more or lesse unequal among themselves and instead of making figure I the triangle CDF of the spaces between the points of shadow CDF of the superficies of the Dyal you shall make it of the spaces between the points that are set in the place of these points of shadow 6 To the Theoriciens MAke in one and the same plain as in the first figure vith three right lines CgkD CrtF DieF a triangle CDF equal and like to the triangle of the three points of shadow fig. IV. CDF make upon the said three straight lines CgkD CrtF DieF three other triangles CBD CbF DBF equal and like to the triangles in the air of the beams of the Sun III. fig. CBD CBF DBF every one to his own By the points B and b I. fig. draw a straight line Bqg that may part in two the Angle DbF. Draw out of the point C at your discretion a straight line aqkty perpendicular to the straight line CgkD and out of the point F draw a straight line hPirx perpendicular to the straight line FeiD make in the triangle Fcb the section or cutting Cl equal with Ca of the triangle CBD and the section Fs with Fh of the triangle FbD from the point t center and space tl draw a bow lm from the point k center and space ka draw a bow am that may meet with the bow lm in m and draw along the straight line km from the point r center and space rs draw a bow sn from the point i center and space i h draw a bow in hn that may meet with the bow sn in n draw along the straight line in Make in the straight line km the section or cutting ku equal with kq. By the point u bring to the straight line aqkty a straight line uy perpendicular to the straight line km make in the straight line in a section iz equal with iP by the point zx carry to the straight line hPirx a straight line zx perpendicular to the straight line in finde out the butt end y common to the two straight lines aqkty and uy And also the butt end x common to the straight lines hPirx and zx draw the straight lines goy and eox find the butt end o common to these straight lines gov eox Make in an other place figure II. a Triangle gqy of the three straight lines as gq gy and yu of the first 7 figure make in the II. fig and in the straight lines gy and gq the section go equal to go of the I. fig. And the section gb also equal to go of the I. figure draw if you will the straight line bo of the second figure Make again in another place fig 3. a Triangle cbo of the three straight lines bo of the Triangle gbo of the second figure And of CB and CO of the first figure and upon bc fig. 3. make the cuts bc bd bf equal to the lines BC BD bF of the first figure every one to his own respectively And if you have done rightly the spaces fo do co of the Triangle cbo fig. 3. are equal with the spaces FO DO CO of the first figure every one to his own respectively Take fig. 3. in the straight line bo according to your discretion the point i other then b make three sticks sharp at both ends and equal to the three spaces ci di si of the third figure mark along upon`the rod or Axeltree the space BI equal to the space bi of the third figure work as I have said and as the fourth figure doth shew you and you shall find the Axeltree of the Dial placed in his right place You may after this manner as in others substitute or bring in other points CDF in stead of those of shadow of the superficies or face of the Dyal and work by this mean every where with the like ease Figure 8 For those that have skill in Geometry THe higher figure is the place of the Dyal with the face unequal to the pin AB and to the three points of shadow CDF all markt as it is said Get a flat and solid thing as a slate a board paceboard or the like Draw upon it in the lower figure a straight line BDFC make in that line three cuts BC BF BD equal with the three spaces BC BF BD of the place of the Dyal each of them to his own respectively then from the point B of the lower figure for the center and from the spaces BC BF BD draw some circles DH FE CG By this means you see whether the spaces BC BD BF of the higher figure or Dyal are equal or unequal one unto another and when these spaces are unequal among themselves as it happens in this example you see which is the least and which is the biggest as in this example the space BD comes to be the shortest of the three Now from the point C of the lower figure for the center and from the space between the two points of shadow C and F of the higher figure draw a circle E that may meet in one point E the circle of the space BF viz. the circle FE for it must meet with it then draw the straight line FB that may go and meet in one point H the circle of the shortest space BD viz. the circle DH Again from the point C of the lower figure for center and space between the two points of shadow C and D of the higher figure draw a circle N that may go and meet in the point N the circle of the shortest space BD viz. the circle DH for it must meet with it From the point F in the lower figure for center and from the space between the two points of shadow FD of the higher figure draw a circle that may meet in the point R the circle of the shortest space BD viz. the circle DH for it must meet with it By this means the three spaces or straight lines DH DR and DN of the circle DH which is that of the shore●t space BD have the conditions that are requisite for the making of a triangle Figure 9 For those that are skilled in Geometry MAke in another place as in the lower figure a triangle DGV of three straight lines equal with the three spaces DH DR DN of the higher figure every one to its own Find in the lower figure the center O of a circle the edge whereof may reach to the points VDG according as the lower figure doth declare Draw a straight line DOE through the Diameter or midd'st of this circle By the point O in the lower figure draw a straight line POQ perpendicular to this Diameter DOE From the point D in the lower figure for the center and space BD of the higher figure draw a circle that may meet as in B the straight line QOP for it must meet with it in one or two points viz. In the times of the

all sorts of people if you come again to these second and third figures you shall know at the very first sight what they mean For the Theoriciens And for those that are skilled in Geometry THe I figure is a plate of some thin flat smooth and solid stuff as Iron tinned or the like being round and having a hole just in the Center greater or lesser according to the occasion The II figure is a straight rod round smooth and solid as of Iron or the like of the bigness of the hole in the plate The III figure is as it were a whirl made of the plate and of the rod put thorow the plate in such sort that it is perpendicular to the said plate as the squire that turns round about doth represent unto you and is so fast that it cannot stir or move In the V figure AB is the peg or pin that hath mark'd unto you the points of the shadow CDF the rods or sticks BC BD BF are solid and strong as of wood or the like having each of them a slope edge in a direct line all along going from the point of the peg B to each point of the shadow CDF and are so turned or ordered that in applying the whirl unto them the edge of the plate may goe andtouch the three slope edges of the rods all at once and the rods or sticks are made fast in this situation in such sort that they cannot move nor stir The rule that crosseth over the three slope edges BC BD BF toucheth them all three or else two at the time only whereby it shews whether those slope edges are all three in one and the same situation or upon one and the same ground or no and on which side is their hollowness when there is any The hand applies the whirl unto it and keeps it there till the Axeltree BO● come to touch the end of the peg or pin B and that at the same time the edge of the plate EDH touch the three slope edges of the rods And when the whirl is placed or setled after this manner the rod is the Axeltree of the Dyal and placed as it ought to be and there remains nothing else but to make it fast in this situation or position The IV figure doth shew that if you goe to make use of thin and supple strings in this practice or working in pulling those two mark'd with Ie and Ih to make them fast in direct lines they would make the two strings mark'd with bc bf to bend so that you can doe nothing exactly with them which is the reason that Monsieur Desargues hath not thought fit to make use of them for the Beams of the Sun but rather of the slope edges of the rods that are both stiff and strong 2 3 To the Theoriciens And others that are skill'd in Geometry THis foregoing figure shews to the eye that all the pieces of the Instrument are made so strong and firm that they cannot bend AB is the pin by whose point B you have had the points of shadow C D F. The three sticks or rods BC BD BF have each of them a slope edge in a direct line at length going from the point of the pin B to the three points of shadow C D F. The slope edges of the two longest sticks or rods BC BF have some portions made in them equal every one to the third and shortest stick BD. The three sticks IH ID IE are every one longer than BD and all three made even then they are joyned all by the end to one of the points EDH of the slope edges of the other sticks BC BD BF and their other ends I are brought together in one and the same point I. The rod BI is straight round smooth and strong as of yron or the like it hath a straight line BI drawn from one end to an other and one of the points B of this line of the said rod toucheth the point of the pin And with an other point I of the same it toucheth the point I of the three rods or sticks This being so the rod BI comes to be the Axeltree of the Dyal rightly placed there remains nothing else but to make it fast in this position or situation The figure shews in the rods that goe from the point of the pin B to the points of shadow CDE how one may make fast those rods at one end to the pin and also all together to one point by binding them to it And how they may be made f●●t at the otherend to one point of the superficies of the Dyal by fastning them to it with mastick plaster cement or thelike This way is more sure than that with the strings But yet it is not the easiest nor the least troublesome in my judgement To the Theoriciens Another resolution of the same kind with the former THe position or placing is given of the four points BC DF and the placing of the two straight lines BE BH that divide in two the angles CBD and DBF and of the two ground plots that passe unto those two straight lines BE and BF and that are perpendicular to the ground plots of those Angles CBD and DBF are given out Therefore the intersection or intercutting of these two ground plots so perpendicular is given But this intercutting is the axeltree of the Dyal therefore the position of the axeltree of the Dyal is given Any one may frame at his pleasure upon that which is granted concerning this composition many other resolutions and divers compositions of problemes and divers general ways of practice In the mean time you shall have here three several ways one after another to see which is the most advantagious for the actual practizing of the Art and to induce you to seek or try if there is any other shorter 4 For the Theoriciens The Composition of the Probleme or Proposition in Consequence of the Resolution made upon the lowermost figure of the first draught THe first figure is the place of the Dyal with the pin and the points of shadow CDF Make a ground plot of it II upon one straight line BD and with one point B three Angles DBN DBR DBH equal to the three Angles of the first figure that are between the beams of the Sun DBC DBF CBF every one to his own respectively From the Center B II figure and from any space BD draw a half circle that may meet in the points DNRH the straight lines BD BN BR BH Make in the third figure a triangle DGV with three spaces equal to the three spaces DH DR DN every one to his respectively as having the condition necessary for that purpose Find the Center O of the circle EVGD drawn about this triangle VGD Draw two Diametters DOE POB of this circle perpendicular one unto another Lengthen one POB sufficiently of one side and on the other From one of the ends D from the other EOD draw as

it is the point of one of the hours of 6 as the point 6 〈◊〉 you shall go on in finding out the points of the other hours which may be found in your Dyal in this manner following Figure 17 To all sorts of People MArk at your discretion in the rule PQ two several points MN and consider the point in the middle of the body of the axeltree close by the stay O that is the Point about which you have turned the string with the corner of the squire You see there three several points unmoveable and fixed viz. the point M and the point N in the middle rule and the point O in the middle of the body of the axeltree rod close to the stay And so having those three points fixed M N O you have by this means the three several distances viz. the measures of the distances that are from one of these three points to the two others viz. the space or distance from the point M to the point N the distance from the point M to the point O and the space from the point N to the point O. Remember two things one is that the point O is in the middle of the body that is to say of the bignesse and not in the out side of the axeltree rod The other is that these two points MN that you have mark'd at discretion in the middle rule are not for all that certainly the points of hour and that they are to serve you to find out the points of hour and perhaps they may chance to be some of them and may be not and perhaps you must blot them out after you have found out the points of hour This being done so take with your Compasse upon the middle rule the distance from the point M to the point N and with this space go to some place that is flat or smooth and set both the feet of your Compasse therein at once as in the figure below in the points M and N and by these two points draw a straight line MN as long at either end as the rule PQ Then go back to the Dyal above take with the Compasse the distance which is from the posnt M to the middle of the bigness of the axeltree close by the stay O or else otherwise take the distance which is from the point M to the axeltree towards the stay O and adde unto it half of the bigness of the Axeltree and with this space MO come back to the figure below set one of the feet of the Compasse to the point M and turning this foot about upon this point M trace with the other foot a line crooked like a bow O go back to the figure above take again with your Compass the distance which is from the point N to the middle of the bigness of the axeltree close by the stay O and with this space come back to the lower figure set one of the feet of the Compass to the point N and turning this foot upon this point N trace with the other foot another crooked line that may meet with the other in one point as O for it must meet with it Then open yout Compass at discretion rather more than lesse and set one of the feet of the Compass so open at discretion to the point O and turning this foot of the Compass upon this point O trace with the other foot a round RGSH Go back to the Dyal in the figure above take with your Compass upon the rule QP the distance which is from one of the points M or N to the point of 6 hours and with this space for example of M6 come back to the lower figure set one of the points or feet of the Compass upon this point M go and mark with the other foot in the line M a point as 6 of the same side upon the rule And so you have in the line MN one and the same thing as you have in the Dyal in the middle rule viz. the three points MN and 6 of the same distance in each of these two straight lines This being done draw in the figure below by the two points O and 6 a straight line O 6 which may divide the round RGSH in two halfs RGS and RHS. Open the Compass at your discretion and as much as the space will give you leave and keeping your Compass so open at discretion set one of the feet to the point S and turning this foot upon this point S trace with the other foot two crooked lines L and D then with the same space remove your Compass out of his place and set one of the feet to the point R and turning this foot about upon this point R trace with the other foot two other crooked lines that may meet in two points L and D the two crooked lines that you have drawn about the point S and mark those two points L and D and draw by those two points a straight line LD which may passe by the point O if you have been exact in your operations So you have divided this round into four quarters of a round with the two straight lines SOR LOD and if the straight line LOD drawn in length comes to meet the line MN in a point as 12 it shews that there is also the point of the hours of 12 in your Dyal viz. in the middle rule between the superficies and the axeltree now divide with your Compass every one of these quarters of the round into six parts equal as you see in the points that are upon the brim of the round RGSH and by the center or middle point of this round O and by every one of the points of these divisions of the edge or brim of the round draw some lines or beams as you see some drawn already that may go and meet the straight line MN as in the points 5 4 3 2 1 11. and these points are the points of the other hours that are to be found in your Dyal 18 17 Figure 18 To all sorts of People NOw take with the Compass in the figure above the space from 6 to 5 and with this space go to the Dyal in the figure below set one of the feet of the Compass to the point 6 and keeping this foot of the Compass upon this point 6 go and mark with the other foot in the middle rule another point 5 and by this means you shall transport with your Compass the space 6 5 ●●rom the line of the figure above which represents your table or the flat place in the Dyal of the figure below upon the middle rule MN so accordingly take with your Compass every one of the other spaces 5 4 4 3 3 2 2 1 1● 12 11 from the higher figure and bring them in this manner to the Dyal upon the middle rule in the lower figure and so you have done in this middle rule in the Dyal of the lower figure all and the same spaces

from the center of the half circle Q go razing or laying even the string 24. VI by making it longer or shorter as need requires as you see in O g mark many several points in the superficies of the Dyall one after an other as for example 24 g VI more or lesse according as the superficies of the Dyall is more or lesse uneven Draw an obscure line by the points 24 g VI and it will be a line of houres after the Italian or Babylonian way and so of all the rest The string h OH shews that you may if need be do the like both of one and of the other side of the center O to go and place of one part or other according to the occasion the line as 2 3 4 5. And if you have a straight line as might be O q which may turn about the center O and be perpendicular to the Axeltree BI and you hold the half circle with this straight line set one at a convenient or reasonable distance from the other And let it be alwayes exactly of the distance of six hours after the French way First of all this string describes the Equinoctial line in the superficies of the Dyall Secondly when one of the two either the half circle or the straight line O q is found in one of the points of the hours of the Equator th' other is likewise found in it in an other point of hour then drawing with a string coming from the center O a straight line that may go from the point as t to the end of the straight line O q which you shall go drawing with this string made shorter or longer as need requires and mark some points of line of hour after the Italian or Babylonian way in the superficies of the Dyall And for this purpose there is nothing so easie as to have a circle of Equator that may be fitted to the half circle and where you may have alwayes a space ready made for it's hour 25 To mark the houres after the Manner of the Iewes 24 for the houres after the Italian way Figure 25 To mark the hours after the manner of the Ancients or the Jewes YOu must know first that it would be very troublesome to draw in the superficies of the Dyall the lines of this kind of hours in such a manner as that they might be alwayes just and right in theory all the year long And therefore it is sufficient to draw them just by demonstration in three points onely viz in their points of both ends and of the middle which are the points of those circles that appear the greatest above the Horizon being parallel to the Equator and of the Equator it self The rest goes as it may and therefore it may be said that the lines of such hours traced in this manner are false in the rest of their length yet Curiositie makes them passe for current Wherefore to mark this kind of lines of hours The higher figure 4 shewes which way you must make this half Circle to turn about viz. about a straight axeltree line placed levell in the center of the Axel-tree of the Dyall And to be short set up and make very fast a rod in a straight line passing to the center O and let it be first within the joynt of the axeltree rod secondly let it be level as the figures do shew of a plummet P. and of a level A this being done tye some strings with a loose knot to this rod so levelled NL as you see NR and LT. Take the string from about the center O stretch it out in a direct or straight line from the center O to one of the points of hour after the French way of the Equinoctial line of the Dyall for example to the point of 1 hour as you see the string OI This string being thus strecht out take the other strings of one or th' other end NL and Crosse over this string OI with them and so go and mark many points in the superficies of the Dyall as TIR Draw an obscure line by those points as TIR it is a line of hours after the manner of the Ancients or the Jews do the like with the other hours and half hours of the Equinoctial line If you leave a rod in the Dyall as NOL the shadow thereof will go and shew these hours continually at length if you will not leave it in the button or center O of the axeltree of the Dyall will shew them 26 for the hight of the sun Figure 26. How to mark the Elevation of the Sun above the Horizon THE higher fig. 3. shews which way you must turn the half circle viz. about a straight axel-line hanging down right Set up your half circle so that it may turn like a weather-cock about a rod hanging down right or plum above or below the axeltree of the Dyal it matters not which VVhilest you turn it thus as it is above said cause in the mean time the string comming from the center O to passe by one of the degrees of the edge of the circle and make the string shorter or longer as need shall require mark with it many several points in the superficies of the Dyal according as you see them rankt one by an other in four places Draw a small or obscure line through all these points and this will be one of the lines of the Elevation of the Sun Count the Degrees in the edge of the circle beginning at the first of the beam which is level and ending at the 90. Beam which is down right or plum Mark in the line of the Dyal the number of the Degrees of the border of the circle where the string passes that hath mark't the points of that line and so of all the others and the shadow of the button of the axeltree which is in the center of the circle will shew the Elevation of the Sum above the Horizon Figure 27 How to mark the Sun rising or East rising of the Sun THE figure 2 above shews how you must place the half circle viz. parallel unto the Horizon I would not put a levell to it to avoid confusion It shews also that one of the Diameters of the circle must be set within the center of the Dyal that is to say thar it must go directly from the south to the North and accordingly the Diameter which is perpendicular to it will go from East to West When your circle is set fast in this position let a plummet op in the lower figure hang from the center O. This being done from each point of Degree of the edge of the circle as from x and from z. mark with a string XT or ZR many points in the superficies of the Dyal Draw a small or obscure line through these points as TY or SR. it is a line of the Suns Eastrising Mark in it the number of Degrees of the point of the circle from whence the string comes according as you will count them to begin either from the East or from the South And so of all the other Degrees accordingly And the shadow of the button O will shew which way the sight of the Sun comes upon the Dyal I will take here occasion to tell you that if for some reason or other you could observe in one and the same day but two shadows of the Sun in stead of three as we have said in the placing of the axeltree in the Dyal the declining of the Sun in that day will serve you for a third shadow or else two other shadows observed in an other day I mean you may find equally the placing of the axeltree by one or other of those ways above mentioned and with 3 shadows and with 2 shadows and the declining of the Sun in that day and with 4 shadows two of one day and two of an other which are three wayes that come all to one 27 for the Eastrising of the sun 28 Figure 28. I do not specifie in this volum these kinds of flat Dyals wherein you may work without knobs or middle rule And where you may draw the Equinoctial line trace out and divide the circle Equator in a word where you may do all yea and in the very superficies of the Dyal you may easily come to know them you self by putting this universall way into practice Here is onely a way how to trace out all the twelue lines of the hours equal after the French way in the flat Dyals where the axeltree meets the superficies athwart in the space that you work in so that you shall have no need of a greater place And what I have already said and what I am now going to say again will serve to find out the way to do the like in all kinds of Dyals universally When you have drawn upon your Dyal the Equinoctial line M 12 M drawn conveniently and divided the circle Equator Q 12 Q bring to the Equinoctial line the beam of the 12 hours Q 12. Draw of both sides of the Equator and from the Equinoctial line a straight line MQ parallel to the beam of 12. hours O 12. bring the beams of the other hours to the first which they shall find of the Equinoctial in rt and of MQ in c d●g Q Bring in the Dyal the line of the twelve hours B. 12. draw by the point M of the Equinoctial line and from the center of the Dyal B a straight line ML parallel to the line of twelve hours B 12 make upon this line ML and upon the point M a triangle LMN like to the triangle in the aire OB 12. and let the angles of these triangles in the points L and B be equal one unto an other Carry the spaces Mq Mg Md Mc from the straight line MQ into the straight line MN viz. from M into N into u into i into o bring by the points N u i o some straight lines NL ub if oh parallel to the side NL of the triangle LMN Carry from the center of the Dyal B by the points r t h f o L Some straight lines BL Bh Bf Bh Bt Br These are such lines of hours as you may continue beyond the center B and mark them according to their orders THE END i.e. That are made without any aim or heed

the ninth cut or that above in the tenth figure in an other place than in that where you had taken it and according to the occasion then bring the sticks to it as before for you may take it anywhere or in any place of the line POQ of one side or other of the point B but the further you can take it from the point B will be better and take it in so many-places that having set the sticks of the points CDF to this point I and mark'ed the space BI upon the Axeltree rod the four points I may at last meet together in one point in the air I. And when the point B of the Axeltree rod is at the point B of the pin AB and when the three ends I of the sticks and the point I of the Axeltree rod are met as you see in the lower figure all four together in one and the same point in the air I. The Axeltree rod will come then to be placed just as it ought to be in the Dyal That if you do not care to be sure that your Dyal must be as just as it is possible for art to do in such a case you may spare one of the four lengths CI DI FI BI and content your self with three only as being sufficient for the Theorie But the fourth will serve you for a proof to see whether or no you have been very exact in working and will justifie the three others Figure 12 To all sorts of People THe figure above shews you how that which you have done with three sticks may be done either with many Compasses with the help of some body or else with other kinds of branches tyed or fastened one with another The same figure above as also the figures below shew how every one of those branches may be of two several pieces which go in by couples into one hoop or ring and slide along one by another and are made fast with a screw to the measure where you will have them to stand upon and these pieces may be made of tinn'd yron or of yron if you are afraid that their points will grow dull by often using them Or otherwise they shew you that insteed of one stick you may have two both sharp at one end which you shall fasten and bind together at the other end of what length or measure you please The same figures do shew you also that two divers branches viz. CI and FI may be fastened together in the place where you will have them to stand together with a presse and a screw to fasten them with The higher figure shews you besides that you may ●●●●en or bind with strings or threds the Axeltree rod with the point B of the pin AB and the two branches CI FI with the Axeltree rod to make them stand fast of themselves in their place When you have found thus the placing of the Axeltree rod it is in your choice either to seal it and fasten it in that place or to place another insteed of it that may go the same way and that may be every way equally distant from it But that you may be the more exact it will be as good to seal or fasten that in the place where the practice of the Draught hath caused it to meet than to place another unlesse there was some occasion or necessity for it 12 Figure 3 To those that have understood what hath been said before HAving understood what I have said before concerning those many wayes of finding the position of the axeltree of the Dyal you may compose others besides making use partly of that of one figure and partly of that of an other For example here is one way composed of two of those that are afore OVt of the third or fifth figure you shall take in the Sun-beams or sticks BC BD BF three spaces equal each unto the other And out of the 5 and 6 figures you shall make a triangle of three lines equal to the three spaces HE DE DH of the third figure and you shall find the center O of the circle circumscribed about this triangle You shall find also within the ground plot of the points HDE the points like to A O of the 6 figure or cut which in this case come to be united together in one and the same point O. That is to say having found one of these two points A O you have found also the other because they are united or gathered together into one So you have in the second figure of the third cut the spaces DO and DI for two sides of a triangle with straight angles or corners ODI whose side DI holds up the straight angle and the sides DO and DI do contain or comprehend it Make this triangle ODI with three sticks or with any other thing that may be strong and small as you will so that you may at your need lengthen the side IO from the right angle O. 3 Set the point D of this triangle DIO to the point of shadow D and holding this point of triangle to this point of shadow D make the side IO of this triangle drawn at length if need be come and touch the point B of the pin AB for if you have been very exact in working it must touch it Take a stick HI of the length of DI set one of the ends to the point H and bring the other end to the point I of the triangle ODI without the side IO leave the end B of the pin AB for it must be so if you have wrought exactly as you ought You may have also an other stick EI of the length DI and set one end to the point E and bring the other likewise to the point I of the triangle DIO without the side OI leave the end B of the pin AB That being done the rod BI comes to be the axeltree of the Dyal and placed as it ought to be and so of all the other wayes that you find besides You may if you will make use of a triangle rectangular EOI. and of the stick HI content your self only with the three equal lengths EI DI HI to find out the point thereby that you may draw from thence a line to the point B without making use of any thing else to know if you have done exactly or no you can not be sure whether you have done well or ill But when you have together with that either a fourth length BI or the straight angle DOI that will serve you to try whether or no you have been exact in your operations for as concerning an effectual execution unlesse you have from time to time such a kind of proof to shew whether you have wrought exactly as you ought you cannot assure your self that your work is as well as it can be done One thing I must tell you that in some certain occasions according to the times and the placing or according as the superficies of

division in 24 and is found to be level gives the point of the hours of 6. The other strings shew you that the way of tracing the points of the other hours is the same as above 20 Figure 21 To all sorts of People VVHen you have brought as I have said by means of this rotunda and the strings all the points of the hours into the superficies of the Dyal You may take away the rotunda if you will and make an end of tracing the lines of the hours as before with the strings and by means of the points of hours which you have brought into the superficies of the Dyal as you see in the figure below the line DELKPQSYIZG And for this purpose by means of the said strings carry a string in a straight line from the axeltree to the point of hour for example I and holding it there stretcht in a straight line carry of one side or other according to the occasion an other string comming also from the axeltree as from I or from B that may go in a straight line as far as the superficies of the Dyal and let it go and touch and crosse over the string IR several times in several places and at every time go with this second string to touch and mark a point in the superficies of the Dyal until you have enough as you see the points D E L K P Q Y I Z G and carry by these points a line sweetned that shall be a line of hour do the like for the lines of the other hours and you have done When you have mark'd in the superficies of the Dyal a point of every one of the hours that are to be found in it If you desire to trace the lines of the hours every one at once without making use of the strings as in the figure above you may do it when it is dark as by night with the light of a candle in that manner as it is exprest in the lower figure Set a light behind the axeltree rod of the Dyal and turn the same lightabout this axeltree untill the shadow of this axeltree come to one of the points of hour I and trace in the superficies of the Dyal a line DELKPQSYIZG all along this shadow of the axeltree that line shall be a line of hour do the like for every one of the other points of hour and you have finish'd your Dyal 22 Instruments to work with all 21 Figure 22 Several Instruments to work withall in these occasions hereafter specified I Did not intend to burden my memory with any thing in this matter but with Monsieur de Sargues universal rules for the placing of the axeltree and for the tracing in a Dyal the hours after the French way without medling with the rest which is more curious than useful But to follow the advice of many considerable persons whom I do honour I have set down also the way to mark that which is commonly called the signs The hours after the Italian or Babylonian way The hours after the manner of the Ancients The elevations of the Sun above the Horizon and the rising of the same And for as much as none can do any of these things universally without using these instruments more or lesse This table following shews to the eye all the pieces that are used in those several occasions These instruments are first a circle a half circle or the quarter of a circle which is all one which is made to turn about its Diameter set fast in it 's due and convenient place or down right as in the fourth figure or level as in the second or third figure or else inclining or hanging downward as in the first figure The way to make this circle to move in all kinds of positions is to set two rings in it's Diameter through which one may put in a stick straight round and smooth about which this circle may turn round like a weather cock about his needle or spear as in the second figure and there must be within those hoop rings a screw to fasten this circle in that place or which way soever you will have it to stand The sticks or rods are represented by the 7th figure with a fork at the end of every one bored in the cheek to put a pin through as you see that one may be set plum or down right and the other level being made fast at one end to the axeltree rod as in the 5th figure And for this purpose also the axeltree rod is bored in O. The 8th figure shews the axeltree rod by it self bored with O and the pin Q put through the hole to shew more plainly that which the 5th figure represents viz. all the pieces set or joyned together being mark'd with qo You see that the hoop rings are near the edge or brim of the circle a purpose to leave the center O and a space about it free having commonly a piece taken off that this circle may turn freely about the forked end that is to say about his center without any let or hinderance at all 23 for the Signes Figure 23 How to mark the Signs GEt figure 2 below a half circle both thin and stiff ctsrd draw there a beam OZS perpendicular to the Diameter CPOQD take on both sides of this beam OS 23 degrees and half for example 23 degrees and half from s towards t and as much from s towards r draw the straight line r t make upon Diameter t r a half circle tzr divide the edge or brim of that half circle in six equal parts as in the points that you see there draw by those points as far as the half circle CtsrU some straight lines that may be perpendicular to the straight line r t bring from the center O by the points that those perpendicular have made upon the edge of the half circle CtsrD some straight lines as you see that the strings shew you and with those lines drawn out sufficiently you shall mark the signs in the Dyal as I shall say You see that the half circle is cut thorow or made hollow from the point P to the point Q round about the center O according to the circumference PZQ which is notch'd also in the points that you see in it which are betwixt every degree of the half circle And the center O and these notches are there a purpose to fasten a string upon them insteed of bringing it from the center O. The two figures s32tgez s45rbcz both on the right and on the left side of the half circle CtsrD shew as you may judge by their letters or coats each of them one half of the figure tsrz of the half circle CtsrD which I have made thus bigger than each half of this figure that one may set in the letters ge cb and some figures 2345 about the edges of the two half circles and also the signs as you see which I could not do in the middle