Selected quad for the lemma: ground_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
ground_n	angle_n	line_n	right_a	2,182	5	9.6733	4	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A06780	A treatise of artificial fire-vvorks both for vvarres and recreation with divers pleasant geometricall obseruations, fortifications, and arithmeticall examples. In fauour of mathematicall students. Newly written in French, and Englished by the authour Tho: [sic] Malthus.; Traité des feux artificiels pour la guerre, et pour la recreation. English	Malthus, Francis.; Cecil, Thomas, fl. 1630, engraver.	1629 (1629)	STC 17217; ESTC S109781	49,979	283

take the depth of a valley TO take the depth of a Valley there is somewhat more difficult then in the former operations because there are more obseruations to bee made and to begin you must from the point B obserue in some place opposite a ●a●ke leuell with the horison as A the leuell whereof you may easilie take at this instrument as is taught in the second proposition by the helpe of the plumbet then from the point B take the distance BC as in the fift proposition or else mecanikelie which being done direct your visuel lines from B to A and to C and leauing the instrument in same state turning your selfe about frame the same angle vpon the plaine which will be FBH then plan● markes vpon the lines BF and BH and vpo● the line BH count as many paces or fathoms as you shall haue found betweene B and C and at the end of your paces 〈…〉 marke which will bee at E thi● being done dispose your instrument to make a right angle placing the crosse vpon the ●irst point of the index for the crosse maketh alwaies a right angle when the crosse is vpon the fift point The instrument being thus disposed walke vpon the line BF vntill you may direct your visuell lines by the extreamities of the crosse to B and E or els to FE which will happen in D and no where else then measure the line DE and it will be equall to the depth required G C the demonstration of this proposition is grounded in the 26 of the 1. of Euclide or vpon the equalite of the two triangles BG C and BDE which are both equall and equall angles Although this operation be somewhat more obscure then the others yet I thinke that it is sufficiently explained and therfore I will goe forward to the vse of the sector PROP. IX The manner how to take either distance or altitude with the Sector SVppose the altitude AD were to be taken to the foot whereof you may approch open the Sector 45 adding to it the sights then going forward or backward vpon the line DI vntill you may see the highest part of the altitude A through the two vpper sights the inferior branch of the Sector being paralell to the earth or horison then measure the distance betweene the center of the Sector and the Tower adding to that distance the length of the leg which supporteth your Sector and that shal be the altitude required as appeares in the figure following AB and BC are of equall distance and adding the length of the foot of the sector you will find that it will be height of BD which doth accomplish the altitude of the Tower But to take a distance in any plaine as in the figure precedent the same operation may be vsed except onely the two branches of the sector shall be turned paralell to the horison or ground hauing first made a right angle at the point G or you may operate otherwise first of all prolong a straight line as EGH of what length you please then open your sector to a right angle and set it the point G so that you may see through two sights the point E and where the line visuel of the other two sights strikes along set vp a merke as in I then goe towards it and at your pleasure in the same line set your sector opening it so that you may see the point E and G and keeping your instrument at the same width in the same place place only turne it to the other side so that you may see the point G thorow the sights which before saw E and where the other visuell line shall cut the first line EGH set a marke which will be in H precisely and that shall be the distance required GE the demonstration of this proposition is grounded vpon the 4. and 26. of the 1. of Euclide PROP. X. How to take any distance or altitude inaccessible with the Sector TO obtaine the altitude AB you must first take the distance BC. as is taught by the 9. Proposition precedent and knowing the distance BC which I suppose to be 100 fathom set in the point C your Sector and direct your visuell line through the two vpper sights to the top of the altitude A and let the branch of the Sector be paralel with the ground then leaving your instrument at the same width let fall a perpendicular line vpon the line of the Sector divided into equall parts passing by the 100 number of the inferior branch noted D and note what nūber the perpendicular doth cut vpon the vpper branch of the Sector noted H which I suppose here to be● the 150 part or number and then set the two points of a paire of compasses vpon the two numbers to wit the one point vpon 100 and the other point vpon 150. of the equall parties then transport the points of the compasse all along one branch of the Sector vpon the line of equall parties and the two points shall denote as many parties vpon the line of the Sector as the Towre doth containe fathoms in altitude adding the length of the foote which supporteth the Sector the demonstration of this Proposition is grounded vpon the 4. Proposition of the 6. of Euclide PROP. XI Of Sines Secants and Tangents BEcause the most noble most artificiall and most certaine way of taking of Altitudes Distances or other Dimentions is by Sines Secants or Tangents I haue set downe their operations in such Propositions as are vsuall in this subiect at the end of this Treatise of Practicall Geometrie and before I enter on the method of operation it is necessary to define what the said Sines Secants Tangents are 1. A righ● Sine is halfe the Subtence of of the double Arke A Subtence sometimes called a Cord is a right line drawne from any part of the Circumference of a Circle vnto any other part of the same Circumference so the right line DE is the Subtence or Cord of the Ark DGH halfe of which is DF the Sine of the Arke DG and so is MN the Subtence of the Circumference or Arke MGN halfe of it is MI which is the Sine of the Arke MG now the Arke MG is halfe of the Arke MGN and MI is halfe of the Subtence of that double Arke viz. MGN hence it is according to the definition aforesaid that the Sine of any Arke is halfe the Subtence of the double Arke by the same reason OT is the Sine of the Arke BO and MS is the Sine of the Arke BOM and so of others Note further that the totall Sine the Sine of 90 or the Radius is nothing else but the Semidiameter of any Circle viz. CB or CG If a line be drawne tou●h a Cir●l● it is called a Tangent a tango as ●he line AB toucheth the Circumference of the Circle in B and so AB is called a Tangent line and is s●●●uated on the Terme of