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
A47183	A supplement to a late treatise, called An essay for the discovery of some new geometrical problems concerning angular sections, resolving what was there problematically proposed; and with some rectification made in the former essay, showing an easie method truly geometrical, without any conick section, or cubick æquation, to sect any angle or arch of a circle into 3. 5. 7. or any other uneven number of equal parts. By G. K.	Keith, George, 1639?-1716.	1697 (1697)	Wing K216A; ESTC R216625	4,362	7

A SUPPLEMENT TO A Late TREATISE CALLED An Essay for the Discovery of some New Geometrical Problems Concerning Angular Sections resolving what was there Problematically proposed and with some Rectification made in the former Essay showing an easie method truly Geometrical without any Conick Section or Cubick Aequation to sect any Angle or Arch of a Circle into 3.5.7 or any other Uneven Number of equal parts By G.K. WHereas it was supposed in the former Proposal that a straight Line could be drawn through the extream Points of three or more Concentrick Arches at both ends the Arches beginning or ending upon a straight Line ●oming from the Center of those Concentrick Arches having equal Cords though not equal Arches Upon further consideration it is found that however seemingly such a Line may appear to be straight in many cases as when the Radius is short or the Angle very acute yet in no case is such a Line mathematically straight but is a regular Curve and can be as regularly drawn and by as true Geometrical Art as any Parabola or other Conick Section can and with greater facility and readiness and which any Tiro who understands nothing of Conick Sections and Cubick Equations may do The way of drawing the said Curve is this Let a short cross-Rule be set at right Angles with another longer Rule and let the length of the cross Rule be at pleasure 2 or 3 Inches or more as 6 or 7 as ye have a mind to make the length of the Cord of each part of the Section of your Angle which as in the following Figure let be 3 Inches and let the just half of the cross-Rule be on the left side of the long Rule and let a small Brass or Steel-Pin be fixed on the right end of the said cross Rule that as the Rule is moved may make an Impression on the Paper as the point of the Compass doth in drawing a Circle The length of the longer Rule is to be as occasion requireth as double or triple the length of the other Having thus prepared your two Rules the one cutting the other at right Angles and the cross-Rule fixed to it though it may be made also moveable on it suppose the Angle given to be trisected is BAC measured by the arch BMC in order to draw the curve Line with one draught of the Hand set the left end of the cross-Rule on the point B and from B let it run or slide along the line BA and as it runs along the said Line let the left side of the long Rule still run through the Center or vertical point A which is most easily done and let it run or slide along from B towards A until the other end of the cross-Rule reach at least to the line AC or further as one pleaseth and the Brass Pin on the other end of the cross Rule shall describe the regular curve FIG Having thus drawn the curve Line at the distance of one half of the cross Rule draw the straight Line DE paralel to AC and where the Curve Line cuts the straight line DE as at I a Line drawn from the center A to I shall by true Geometry trisect the given angle BAC The Demonstration Seeing it is the property of these two Rules crossing each other at right Angles where ever the two ends of the cross Rule terminates to make an Isosceles triangle making always two right angle triangles whose Bases are equal and the Perpendicular common to both therefore by the 4. 1. el. Eucl. the Hypotenusals are equal Therefore with Radius AI describing the arch HLIS draw the Cord HI and from I let fall a perpendicular on the line AC as IK making HI = LI = IK therefore the arches of those equal Sines are equal as HL = LI = IS q.e.d. The same or any other Angle obtruse or acute may be trisected into 3 equal parts without the curve Line or any part of it by finding the point I which can be found without the curve line by letting the cross Rule slide or run along the line AB while the left side of the long Rule still runneth through the center A either upwards or downwards until the right side of the cross Rule touch the straight line DE which shall be at I. And thus without any need of noticing or regarding the Curve Line the Point 1 is found where the two straight lines HI and DE meet together And as thus any Angle may be trisected without drawing any Curve Line so it may easily and truly be done without either Scale or Compass other than what the two cross Rules are as any Artist may easily perceive If any object against this Method as Mechanical and not Mathematical and truly Geometrical because performed by an Instrument I shall refer them to two great Geometricians for its Vindication to wit Des Cartes in his second Book of Geometry and Franciscus a Schoten in his Commentary on him argum lib. 2. both which do prove that what is performed by Instruments Geometrically made is Geometrical otherwise the plainest Geometry must be rejected because its Figures are drawn by Rule and Compass both which are Instruments and not only Parabolas and other Conick Sections which are Curves but divers other Curves yea all such that can be drawn by Art with the help of Instruments such as they have devised they contend to be truly Geometrical and both of them in their Geometrical Treatises use divers Instruments for describing Curves Geometrically much more difficult to be made and with more difficulty to be used than what is here proposed of two simple Rules cutting one another at right Angles And seeing it hath no dependance on Solids or Algebra Equations and may be done without any Curve Line as is above showed and whose demonstration wholly depends on a few easie Propositions of the first Book of Euclid I see not why it may not be called Plain Geometry And as the word Mechanical is used to signifie a thing not Mathematically exact but coming near to it by Approximation in this sense it is not Mechanical but Mathematical and purely Geometrical being grounded on as good demonstration as any Propositions in Euclid and being but a Corrolary from some of them The next thing to be shown is the Quinquisection where to make one Figure serve to both I make the Cross Rule only one Inch and one 4th part from the middle line AM setting off on both sides one half of the length of the cross Rule draw the paralel Lines ad and be then let the cross Rule side along the line AB as in the Trisection while the left side of the long Rule slides through the center A the other end of the cross Rule shall describe a Curve a part of which shall be g h that may be continued at pleasure Again setting the right end of the cross Rule one the Point d let it slide or move along the Line da