Selected quad for the lemma: book_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
book_n	demonstration_n	proposition_n	quantity_n	1,827	5	13.9195	5	false

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

while the left side of the long Rule runneth through the center A the left end of the cross Rule shall describe a part of another Curve meeting at h the other Curve And having found the point h with radius A h describe the Arch v h z x y N which shall give v h = one fifth of the whole Arch as is evident from the foregoing demonstration The Quinquisection also may be made without any Curve if two long Rules be jointed together like a Sector and each have a moveable cross Rule to move on them at right Angles with the long Rules For let the center of the two long Rules be fixed on the center A and let the 2 cross Rules be moved together from B and d until the left end of the one still touching the right end of the other the right end of the Cross nearest to the Line a d touch upon some Point of it as at W the Point at W shall give the Quinquisection as above And thus a true Geometrical Line of Cords may be made by any Tiro without any Conick Section or Algebra Equation and without any Table of Natural Sines or Arithmetical Operation for whereas Euclid 11.4 hath taught how to find the Cord of 36 degr and also it is found by Quinquisecting the half Circle as is above shewed it remains only to trisect the Arch of 120 degr which giveth the Cord of 40 and 36 taken from 40 leaveth 4 degr which bisected gives 2 and that bisected giveth 1 which is the one 360th part of the Circle and one 90th of the Quadrant and this is more methodical than to teach a beginner to make his Line of Cords for projecting of Angles by sending him to Conick Sections and Algebra Equations or the Table of Natural Sines which he is not capable at his entry nor after he has made some good progress to understand it being to teach ignotum per ignotius an unknown thing by a more unknown quite contrary to all good method of true Science such as Geometry is The method of the Quinquisection here delivered sufficiently showeth without example any other Section desired The Corrolaries mentioned in the former Treatise with the Rectification here made are all valid some of the chief of them I shall here mention 1. One great Use is to teach a beginner how to make a true Line of Cords as is above showed and how to divide a Circle into any parts required 2. Another great use Descartes showeth in his third Book of Geometry for the resolving any such Equation in Algebra as z 3 = + p zq where the Root z is an unknown quantity and can be found by the Trisection of an Angle 3. A third great use is to give some New Promblems in Practical Geometry one whereof I shall here show Let a straight Line A F be given see the second Figure and it is required on the point A to erect an Isosceles ABC whose side BC produced shall terminate on a limited point D under the given straight line AF. The construction is thus draw a straight Line from D to A as D A next make the right Angle FAE Divide the angle EAD into three equal parts and with radius AD describe the Semicircle GFE From C to B set off GB = ED. Then draw the line AB and from B draw the line BCD which shall form the Isosceles triangle ABC whose side BC being produced shall terminate on D. q.e.f. the use of this is obvious in Architecture 4. A fourth great use is to give us some New Problems in Geography and Navigation Example There are four places A. B C. D so situated A is distant from D 100 Leagues and beareth South-Easterly from it 70 degr B is distant from A 100 Leagues South-Westerly C is distant from B 100 Leagues North-Westerly C and A are in the same Latitude and so that these three places B. C. D lie in a straight Line one from another Q. What is the distance betwixt these two places B and D and the Course on the Rhumb Line betwixt A and B and the distance betwixt A and C. The Resolution Divide the angle EAD into three equal parts and make CAB = one third part of EAD and draw the line BAD Thus the four places A. B. C. D shall be duly situated and an Isosceles Triangle shall be formed ABD whose side AB = AD = 100 Leagues and the angle BAD = 86 degr 40. consequently by plain Trigonometry the angle of the course GAB being found which is 23 degr 30 min. the angle ABD its double is 46 degr 40 = ADB by the Rule of Opposits As the Sine of 46 d. 40 to AD 100 Leagues so the Sine of 86. 40 Log. 9.861757 2.000000 9.999265 11.999265 9.861757 to BD 137. ● fere 2.137508 From which substracting BC 100 Leagues there remaineth CD 37. ● as was required A Fifth great use of the Trisection and other Sections is having the Ratio of any 2 Angles given in any plain Triangle to find the quantity of them if the quantity of the 3d Angle be given without any regard to their sides What other uses these Angular Sections may have is left to the search of Industrious Artists London Printed for the Author and are to be had at the Three Pigeons over against the Exchange and at his House in Pudding-lane at the Sign of the Golden Ball where he Teacheth the Mathematical Arts.