Linguistic DNA - Concept Modelling Demonstrator

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	let_v	line_n	perpendicular_a	1,964	5	14.8700	5	true

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A31232	The English globe being a stabil and immobil one, performing what the ordinary globes do, and much more / invented and described by the Right Honorable, the Earl of Castlemaine ; and now publish't by Joseph Moxon ...	Castlemaine, Roger Palmer, Earl of, 1634-1705.; Moxon, Joseph, 1627-1691.	1679 (1679)	Wing C1242; ESTC R3787	128,382	169

shall here begin I. Upon a Line given AB to erect CD a Perpendicular IF there be a Point as C given in AB the Line on which the Perpendicular is to fall Mark on both sides of the said Point with your Compass the equidistant Points M and N then opening them at pleasure put one foot on M and describe the blind Arch EF and putting the other Foot in N describe the blind Arch GH and the fair line from D their Intersection to the Point C will be the Perpendicular requir'd Now if you have no Point assign'd in the said Line AB to terminate your Perpendicular by take two Points there at pleasure as suppose M and N and opening how you will your Compasses describe the blind Arches EF and GH above your Line and OP and QR below it and the Intersections of these Arches to wit D and S will be two points to draw your Perpendicular by II. Upon C the end of AC a given Line to draw DC a Perpendicular OPEN your Compasses at a convenient width and putting one Foot on C let the other within reach of AC mark any where as at F then touching or cutting from thence the said AC with the moving Foot of your Compasses at suppose E and describing on the other side of F the blind Arch GH lay your Ruler on FE and it will cut the said Arch at suppose D so that DC will be the requir'd Perpendicular III. A Line AB being given how to draw DG a Parallel to it HAVING taken two points in the said Line as suppose A and B open your Compasses at what width you please and putting one foot on A describe the blind Arch CDE and putting one foot on B describe the blind Arch FGH then if you lay your Ruler on the highest part or greatest Extuberancy of the said Arches to wit on the Points D and G the Line so drawn will be the requir'd Parallel IV. To describe a true Square AB being a Line as long as the side of the Square you design erect on the end A the Perpendicular DA of the former length then taking between your Compasses the said AB put one foot on D and describe the blind arch EF and again putting one foot on B describe the blind arch GH to cut EF and if from their Intersection C you draw the fair lines CB and CD you have a true Square V. To draw an Oblong or as they commonby call it a Long Square AB being the longest side of this Square erect on the end A the Perpedicular DA of the length of the shortest then taking between your Compasses the line AB put one foot on D and describe the blind arch EF and taking between your Compasses the line AD describe the blind arch GH to cut the said EF and if from their Intersection C you draw the fair lines CB and CD you have the Square you design VI. To Describe an equilateral Triangle or an Isosceles OPEN your Compasses at AB being the side of the Triangle you design and putting one foot on A describe the blind Arch EF and again putting one foot on B describe the blind Arch GH to cut the said EF and if from their Intersection C you draw the fair lines CA and CB you have a true equilateral Triangle Nor is there any difference in the Description of the Isosceles ASB for the only difference between them is that the sides AS and BS of the Isosceles are longer or if you please they may be shorter than the Base AB whenas all three sides are equal in the equilateral Triangle VII To make a Triangle of three given Lines SUPPOSE the first line given be AB the second AC the third BC and that you are to make a Triangle of them let AB be the Base and taking the given line AC between your Compasses put one foot on the Base at A and describe the Blind Arch EF then taking the given line BC between your compasses put one foot on the Base at B and describe the Blind Arch GH to cut the said Arch EF and if you draw lines from their Intersection at C to A and B on the aforesaid Base you have your intent VIII To describe an Oval CROSS RP at right Angles with IM and taking with your Compasses on the said lines from the intersection O equal distances to wit OA OB OC and OD and draw through the point C the lines AK and BH each equal to twice AC as also throu ' D the lines AN and BL each equal to twice BC then A and B being Centers describe the Arches KPM and HLR in like manner C and D being Centers describe the Arches HIK and LMN and the figure thus drawn will be a perfect Oval So much for the Geometrical Problems necessary for Dialling and as for the Instrumental ones i. e. those performed by the Sector they are as I may say of two sorts some belonging to one side of it and some to the other for the side marked with L is divided into 100 equal parts and called the LINE of LINES and the side mark'd with S the LINE of SINES First then of the LINE of LINES which by the way tho' it be divided as I said but into 100 parts may yet stand for 1000 if you fancy every 10 Divisions a Line of 100 parts and in like manner it will stand for 10000 parts if every division be deemed 100 therefore a Line v. g. of 75 equal parts may be exprest by 75 of those Divisions or by 7Â½ or by Â¾ The Use of the LINE of LINES marked with L. I. To divide a Line into any number of equal parts SUPPOSE your Line were to be divided in 23 equal parts take it between your Compasses and opening your Sector place one foot of your said Compasses on the 23 division of the Sector and the other foot on the 23 over against it and the distance between the Figures 1 and 1 on the said Sector will give you one equal Division of your Line and the distance between 2 and 2 will give you two equal Divisions of it and in this manner proceed till you quite run over it as you design II. To find the proportion between any two Lines SET over the greater Line at 100 and 100 on the Sector then taking the lesser between your Compasses find where it will be just set over also or lye parallel to the former which hapning suppose at 50 and 50 you may conclude that the Proportion required is as 100 to 50. III. To divide a Line as any other Line proposed is divided that is to say according to any Proportion SUppose you saw a Line containing 65 equal parts of the Sector devided into three pieces the first containing five equal parts of the Sector the other fifteen so that the last must be 45 then suppose you would divide after this proportion another Line containing but thirteen equal

posture of all the Stars so that if those you seek after be near the Horizon Meridian or any other noted Quarter those on your Plane near its Horizon Meridian or corresponding Quarter will resolve the Question Or if you take the height of a Star and its Azimuth according to any of the former Directions then whatever Star on your Plane has the same it will be that you seek after and consequently you have its Name Now knowing once a Star your said first Plane shows you what they are that lye about it and so by degrees you may run from one to another round the Heavens Nor need you as to the knowing of the Stars be so exact either in rectifying your Projection or in having the hour of the night or in taking the Heights and the like as in other Operations for by the bigness of the Star by its nearness to some remarkable one and by twenty other particular properties you will be so regulated and confined that you may safely conclude when you examine your Projection that the real Star you see can be no other than such and such a one How to Describe the PROJECTION HAving thus show'd you the use of the Pedestal or Projection I shall fall on the way of Describing it and according to my manner all along on the Demonstration of it also especially since it conduces to a more easy comprehension of all Steriographical Projections and if I be a little longer than ordinary it is now no great matter for I have ended all the Operations I intend at present so that what is here further said may be omitted without inconvenience if the Reader be disgusted at Speculation As for the nature of the Projection t is Optical representing all things in the Heavens as they appear to the Eye from such and such a Station and not according to their true and real distances 'T is chiefly founded on the 20th Proposition of the third Book of Euclid which proves that the Angle at the Periphery is but Â½ that at the Center for from thence 't is infer'd that if placeing our Eye on the superficies of the Sphere v. g. at the South Pole we look into its Cavity the Angle made at our Eye by the two Rayes that issue from it the one along or throu ' the Axis to the opposite Pole and the other to a determined Point will be the Angle only of half the value of the Arch or real distance between the two Objects i. e. between the said Opposite Pole and Point now since any Diameter on the Plane of the Aequator for that or some Parallel Circle to it we now suppose to be the Plane of our present Projection meeting with those Rays will be the Tangent of the Angle they make which being in value as we said but half the real distance between the said Objects it must need follow if any Star or Point in the Heavens be distant from this opposite Pole suppose 20. Degrees That the Tangent of 10 Degrees from the Center of the Projection which represents the said Pole gives its true apparrent place there and the like is to be said of any other distance I shall not trouble the Reader with any Scheme to demonstrate this further because being fusely treated of by Aguilonius and others 't is obvious enough to all Mathematicians and as for new Beginners if they desire a fuller conception of it let them but apply themselves to any man vers't in Projections and in the space of ten Minutes he will shew it them more clearly and naturally by Strings fitly placed on an Armillary Sphere than I can here in many hours therefore supposing if to such what I have already said be not evident that the Heavens may be thus projected by half Tangents let us proceed to the way of doing it that is to say to the finding of the Centers and Radius's of all the Circles which conduce to the before mentioned Operations As for the Concentric Circles of the first Plane to wit the Aequator the Tropics and the Limb which is as I said Circulus maximus semper latentium or some Parallel to it there is no difficulty in describing them for having drawn at right Angles the Lines NS and EW representing the four Cardinal Points throu'P the Center or projected Pole if you open your Compasses at the Tangent of 45 Degrees and place one foot on the said P you must needs project the Aequator because being distant from either Pole 90 Degrees the Ray that touches it and that which runs along the Axis to the opposite or North Pole makes an Angle at your Eye as we said before of only half so much In like manner the Tropic of Cancer being 66 g. 30 m. from this Pole the Tangent of 33 g. 15 m. gives his Radius as the Tangent of 56. g. 45 m. does Capricorn whose real distance from the said Pole is 113 g. 30 m. for it lies 47 Degrees beyond the former Tropick And lastly the Tangent of 64 g. 15 m. projects the Limb or uttermost Circle if it be Circulus maximus super latentium as being yet 15 Degrees further for the true distance of that Circle from the said Pole 128 Degrees and 30 Minutes Now for the main matter to wit the great Circles which fall obliquely on the Plane take this easy general Rule for them all viz. That their Centers are distant from the Center of the Projection the Tangent of as many Degrees as their Poles are distant from the Pole of the Plane on which the Projection is made that is to say in our present Case from the North-Pole of the World and the Secant of the said Degrees is their Radius Suppose then you were to project v. g. the Ecliptic which is the only oblique Circle of your first Plane you know that its Northern Pole being in your Meridian is distant from the North Pole of the World 23 g. 30 m. Open therefore your Compasses at the Tangent of those Degrees and place one Foot in P and the other will give you in the Line PN the Northern half of the Meridian of your Plane or in the Line PS the Southern half of the said Meridian the point D for the requir'd Center D then being the Center open but your Compasses at the Secant of the said Degrees and you have the Radius Nay the Distance from D to e or from D to w the East and West Points of the Aequator or points where the Ecliptic intersects with the Aequator on the Sphere gives this Secant for if PD be the Tangent of 23 g. 30 m. then D e and D w are you see the Secants But before we demonstrate the aforesaid Rule let us make an end with the great Oblique Circles of the Transparent or second Plane which are only the Horizon HRST and the Azimuths of every 10 Degrees exprest as I said by plain Pricks and

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