Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	divide_v	equal_a	line_n	1,364	5	9.1106	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A61369	A compendium of fortification, both geometrically and instrumentally, by a scale the making whereof is shewed by the tables, and their use, both of the tables and the scale, for speedy protracting of any fort consisting of 8 bulwarks, whose bastion-angles shall not exceed 90 degrees, and so the like for bastion-angles of 12 bulwarks / written by Philip Staynred ...	Staynred, Philip.	1669 (1669)	Wing S5357; ESTC R23470	7,877	12

seventh Figure Now to reduce it from the Bastion Point at A you must take A B the shortest side and lay it from O unto Q and from Q draw a Line parallel to the Capital Line K N continued as Q R. Lastly drawing a Line from N to O it shall cut Q R in the Point S so is Q S the length of the Capital Line sought for which must be laid down on the seventh Figure from A unto T so is T the Center of the Gorge Then for the Front take K V and lay it on the Capital Line from K to W so a Ruler being laid from W to O it shall cut the Line Q R in the Point at X so is Q X the length of the Front to be laid down in the seventh Figure from A unto H and I. Thus shall you finish your Bastion when you have let fallen your Flanks perpendicular on the ends of your Curtains as you see The like Method you are to observe for the other Bastions And when you have finished your Fort you must observe whether your Curtain Lines that is from the Centers of the Gorge be parallel to the outward Sides A B c. which if they are not you must correct them and by your Judgment by help of the Lines of Defence you may as you see occasion widen the Necks of the Gorges and also the Bastion Angles but not above 90 Degrees And so let the Flankers be as neer proportional as the Rules or Ocular Demonstration directeth which commonly the Gorge Line to the Flank bears proportion as 7 to 6. Much more could I write of Irregular Fortification but my purpose at this time is but to make a small Treatise or an Epitome thereof The seventh Figure of an Irregular Fort containing 5 Bastions being the Platform of the Royal Fort sometimes on St. Michael's Hill on the North West Side of the City of Bristol To make a Scale for Fortification by the Tables THis may be performed Geometrically by observing the former Instructions whereby you may gain the length of every Line but it will be sooner done and more easie by these Tables following First Table for 12 Sides Numb of Sides 4 5 6 7 8 9 10 11 12 Semi Out Pol. 1000 1000 1000 1000 1000 1000 1000 1000 1000 Semi Inn. Pol. 661 700 731 756 777 795 810 823 834 The Front 412 347 299 261 232 209 190 174 160 The Gorge 158 151 141 132 123 115 108 101 96 The Flank 133 127 119 111 103 97 90 85 80 Second Table for 8 Sides Number of Sides 4 5 6 7 8 Semi Outer Polyg 1000 1000 1000 1000 1000 Semi Inner Polyg 661 706 740 766 787 The Front 412 346 296 259 229 The Gorge 158 156 148 138 130 The Flank 133 131 124 116 109 Make your Scale of a sufficient length that may hold both Lines the one for 12 Sides and the other for 8. Make choice within the breadth of the Scale between the Borders any sufficient breadth as C D from whence draw Parallels to the Sides and divide C D into 30 Equal parts and begin your Account from C with 45 so shall the end at D be 75 degr Now make choice of the length of the outer Polygon which here I make three Inches and divide a Line by the Side thereof equal thereto in 100 equal parts and suppose each part into 10 so have you 1000 parts agreeable to the Tables The next thing is to draw Parallels to C E according to the Polygon half Angles as you may see in the Tables under the Pentagon Fort being the second Figure So from the Scale C D you have for the half Angle of a Pentagon 54 Degrees whereby you may draw the Pentagon parallel F G and so in the lower Scale of 8 Bastions In the like manner you may do for all the rest Now to draw the cross Lines for the Semidiameters of the Inner Polygons as also the Lines for the Fronts Gorges and Flanks you shall work thus First you must note That the Semidiameter of the Outer Polygon is the Radius or whole Line of 1000 equal parts and that is drawn at Right Angles or a cross at F But for the Semidiameter of the Inner Polygon look in the Table of 12 Sides in the second Column under 4 you have 661. Take the same Number off your Scale of Equal Parts and lay it from E to H Then in the third Column under 5 you have 700 parts lay the same down from G to I and make there a prick or point Do the like for the Hexagon and Heptagon as at I and K proceeding along with all the rest unto the Dodecagon And lastly draw a Line through all those Points So have you the Arch Line H M for the Semidiameters of the Inner Polygon In the same manner work for the Front Gorge and Flank Lines The Scale of 8 Sides is the same Method I have also inserted on the left side of the Scale a Line of Chords whose Radius or Arch of 60 Degrees is three Inches and on the left side a Line for the Sides of the Polygons The Hexagon or six Sides is equal to the Radius And for the Tetragon or four Sides it is equal to the Chord of 90 Degrees So having described a whole Circle with the Chord of 60 Degrees you shall find that if you take from the Center N unto 5 it shall divide the Circle into 5 equal parts for drawing the Figure of a Pentagon which in the second Figure of a Pentagon Fort will reach from A to B and so to X Y Z and A. Now D A in the same second Figure you shall find to be the Semidiameter of the Outer Polygon which in the Scale is G F and taking G I off the Scale for 12 Bastions or G O on the Scale of 8 Bastions it will give the Semidiameter of the Inner Polygon as D E in the same Figure So likewise G P on the Scale will be equal to the Front A K in the Pentagon Fort. The like you may understand for laying down the Gorge and the Flank And for the Curtain as before you must make R S 1 ½ the length of the Front A K. This Scale will also be of good Use in Irregular Fortification As for Instance In the Irregular Fort the seventh Figure you shall find the half Bastion Angle G A F to be 58 Degrees which falls on the Scale between the Pentagon and Hexagon from whence you may draw a Bastion on some spare place and from thence proportionate the same unto the outer side of the seventh Figure A B. The rest I leave to your own practice How to Fortifie a long Curtain with Bulwarks or a strait Town Wall LEt the Curtain be A B. First take 200 Foot from the Scale of Fortification accounting 10 for 100 and lay the same from A unto C and from C unto D So shall A D be the breadth of the Neck of the Gorge and upon the Point C erect a Perpendicular as C F. Then take 420 Foot off your Scale and lay the same from D to E which shall be the length of the Curtain Next you must take 720 Foot and lay the same from E to cut off the Capital Line at F so shall E F be the longest Line of Defence and C F the Capital Line which Line C F must be laid down from C unto G and draw G F for the shortest Line of Defence Lastly upon D erect a Perpendicular which will cut the same Line in H so have you D H for the Flank and H F the Front Thus have you finished half the Bastion from whence you may transport all the rest of the Bastions were they ever so many Note That these Bastions must not exceed 720 Foot that so it may not be without Musket-shot But if you will defend the Front with Cannon you may make the Line of Defence almost twice so much The like for the Curtain which may be 800 Foot and in the middle of the Curtain you may make a Spur or Point of a Bastion as at K which will be necessary for Musket-shot beside the Cannon which in the Line drawn about the City of Bristol I have seen many of them These five following Pieces are taken out of Malthus The Proportions you may find by the Scale and the Rules before shewed Tetragon Pentagon Hexagon Heptagon octagon FINIS