Selected quad for the lemma: order_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
order_n	double_a	open_a	rank_n	7,184	5	12.3481	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A64224	Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.	Taylor, John, mathematician.	1687 (1687)	Wing T534; ESTC R23734	190,995	514

an Equilaterial Triangle PPP Now divide PP into three parts then take 1 and prick off the Capitals PA c. and the Gorges make equal thereunto as PC PC c. then make the Flanks FC to stand at Right Angles and to be ½ of PC or PA then draw the Faces AF AF c. and the Work is finished as required PROP. X. To fortifie a Square with Demi-Bastions The sides of the Square may be from 100 to 200 Feet let PP be 180 Feet which divide into 3 parts take one for the Gorges PC and for the Capitals PA and prick them off all round as you see then take ⅙ of PP and at Right Angles prick off the Flanks CF then draw the Faces AF AF c. and the Figure is compleated PROP. XI To fortifie a Parallelogram with Demi-Bastions and Tong. First describe the Parallelogram or LongSquare PPPP then divide PP into 6 parts the side on which the Tong or Tenaile is placed and make MC equal unto ⅙ thereof and also MG and MH Draw CG GC and CH HC then finish the Demi-Bastions as before so shall the Work be compleated as was required A Long Square may also be fortified as Fig. the 77. PROP. XII To fortifie a Star Redoubt of 4 5 or 6 Points 1. A Star Redoubt of four points may have his side from 40 to 60 Feet First describe the Square PPPP then divide PP into two parts at M take ¼ of PM and by prop. 1. § 1. ch 4. raise Perpendiculars round at M make MA equal to ¼ of PM and draw all as in the Figure 2. A Star Redoubt of five points is thus fortified Describe the Pentagon PP c. then divide PP into halves at M raise the Perpendiculars MA make MA equal to ⅓ of PM and draw the Fort in all respects as the Figure representeth 3. A Star Redoubt of six points is thus fortified Describe the Hexagon PPP c. divide PP into two equal parts at M then raise Perpendiculars at the M' s then make MA equal to ½ of PM or ¼ of PP and draw every respective line as you see in the Figure PROP. XIII To Delineate a Plain Redoubt Plain Redoubts are called Grand Redoubts which are used as Batteries in Approaches whose side may be from 60 to 80 Feet or Petit Redoubt which are used for a Court of Guards in the Trenches and may be from 20 to 50 Feet and are framed and delineated in all respects as you see in Fig 90. The Profile's to be set on these several Works and the Motes are alterable and uncertain for they being sometimes used in Approaches then they do require the Breast-work at the Bottom to be 7 or 8 Foot wide and the Interior Height 6 and the Exterior 5 Feet and the Mote to be either 8 10 or 12 Feet sometimes 14 or 20 Feet wide at the bottom and the height of 7 8 or 9 Feet to have two or three ascents to rise to the Parapet There are many other things belonging to this Art which the limitation I am bound to will not permit here to be treated of CHAP. XIV Of Military Orders or the Embattelling and Encamping of Souldiers SECT I. Of the Embattelling and Ordering of Souldiers BATTAILS are considered either in respect of the number of Men or in respect of the form of Ground In the respect of the number of Men it is either a Square Battail a Double Battail a Battail of the Grand Front or a Battail of any proportion of the number in Rank to the number in File In respect of the form of the Ground the Battail is either a Geometrical Square of Ground or a long Square of Ground For the Distance or Order of Souldiers martialled in Array is distinguished either into Open Order which is when the Centers of their places are 7 Feet distant assuunder both in Rank and File or Order which is when the Centers of the places are 3 ½ Feet distant both in Rank and File or else 3 ½ Feet in Rank and 7 Feet in File PROP. I. To Order any number of Souldiers into a Square Battail of Men. Admit it were required to Martial into a Square Battail 16129 Men To doe which extract the Square Root of 16129 by prop. 8. § 1. chap. 1. which is 127 therefore you are to place 127 Men in Rank and also in File PROP. II. To Order any number of Souldiers into a Double Battail Admit 16928 Men were to be Martialled into a Double Battail extract the Root of half the number of Men i. e. of 8464 whose Root is 92 therefore I say that 92 Men must be placed in File and 184 in Rank to order that number of Men propounded into a Double Battail Plate VII Page 302 PROP. III. To Order any number of Souldiers into a Battail of the Grand Front. Admit 16900 Souldiers were to be Martialled into a Battail of the Grand Front that is Quadruple Extract the Square Root of 4225 that is ¼ of the Men the Root is 65 therefore I say 65 must be placed in File and 260 in Rank to form a Battail of the Grand Front. PROP. IV. Any number of Men together with their distance in Rank and File being propounded to Order them into a Square Battail of Ground Admit 2500 Souldiers were to be Martialled into a Square Battail of Ground in such sort that their distance in File should be 7 feet and in Rank 3 feet and 't is required to know how many Men must be placed in Rank and in File to draw up 2500 Men into Square Battail of the Ground According to prop. 1. § 1. ch 1. say As 7 to 3 So is 2500 to 1071 c. whose Square Root is 32 c. Therefore I say 32 Men are to be placed in File Now to find how many Men are to be placed in Rank divide 2500 by 32 the Quotient is 78 which are the number of Men to be placed in Rank and 4 Men to be disposed elsewhere PROP. V. Any number of Souldiers propounded to Order them in Rank and File according to the reason of any two Numbers given Admit 6400 Souldiers are to be Martialled into Array in such Order that the number of Men placed in File shall bear such proportion to the number in Rank as 7 to 13 according to prop. 1. § 1. chap. 1. say as 7 to 13 so is 6400 to 11885 c. whose Square Root is 109 c. the number of Men to be placed in Rank by which divide 6400 it produces 58 c. the number of Men to be placed in File and 78 Men to be employed elsewhere SECT II. Of Castermetation or Quartering and Encamping of Souldiers IN Quartering and Encamping of Souldiers it is requisite the Quarter-Master General and all other under Quarter-Masters be skilled at Foot measure that so they may lay out their Quarters as directed The common allowance for the depth of Ground that a

Cazemat be 6 Feet as before and let all the rest be compleated as you see in the Figure SECT VI. Of the Dimensions and Constructions of those Out-Works called Ravelins Horn Crown-works c. THE Ravelin is a certain Work lying beyond the Mote or Ditch for the covering the Curtain Bridge and Gate the Angle of the Ravelin must not be less than 60 nor much above 100 deg the manner of delineating it is thus Lengthen out the middle line of the Curtain OM unto a convenient length then take with your Compasses the length of the Curtain CC and setting on Foot in F the point of the Face and Flank cross the middle line in q then laying your Ruler at q and to the points F draw the lines of the Ravelin q R and qS which shall be the Ground-lines of the Ravelin The M●te surrounding it must be half the breadth of the Great Mote the Rampert may be 30 Feet thick and some 6 7 or 8 Feet high on which may stand a Parapet equal to that of the Rampire Now if from the points F you raise streight lines into the Campaign at Right Angles to the Curtain and from the points F set off FE and FE 720 or 750 Feet then may you joyn EE either with A Single Tenaile which is done by joyning EE and dividing it into four equal parts take one and place from D to N and so draw EN and EN so have you a Single Tenail IENEL which must have a Mote Rampire and Parapet like the Ravelin Secondly it may if occasion require be fortified with A Horn-Work which is done by joyning the points E E and fortifying the Exterior Polygon EE as is afore taught Or divide EE into three parts make ME and EN equal to MO then draw N M which divide so likewise at O and P then draw E O and E P then at P and O raise Perpendiculars O Q and P R so shall M E Q O P R E N be the Horn-Work which was desired which must likewise have small Rampires and Parapets as afore For the Crown-Work From the Center of the Fort O draw O M B of a convenient length then from the middle of the Ravelin set off 1000 or 800 Feet to B then on q strike the Arch D B E set off the Curtain and Demi-Gorge P C C from B to F and G both ways then draw C F and C E to terminate the points I and H on the Counterscerp then toke ⅓ part of B F or B E and set it from B to M and srom F to L and from E to M then draw L M and M N then for your Demi-Bastions make N P and L O equal to N E c. Then for the Demi-Gorges of the whole Bastion in the middle let them be equal to ⅕ of the Interiour Poligon L M or M N viz. M Y or M X then finish the Bastions by drawing the lines of Defence and raising Perpendiculars or making Angles of 98° at O X Y and P then the Crown work is finished as desired You may make Ravelins and other Works beforementioned before these Curtains if occasion require There are some other Works which are used as Half Moons Bonnets Double Tenails Counter-gards Horseshoes Priests-Caps c. which would be superfluous to speak of in this place 5. Cittadels are Castles or Forts of the least sort and are the Out-works lastly used which are commonly of 4 or 5 Bastions and are placed in such Order that there may be two Faces and a Curtain towards the Town the Construction whereof is after this manner Lengthen out the line OM and therein find the Center of the Cittadel the Interior Polygon of the Pentagon may be ¾ of the Curtain adjoyning or a little more the Center of the Square may be on P the point of the Interior Polygon the Center of the Hexagon may be near the outward point of the Bastion of the Town taken away to make the Cittadel in which may be delineated as afore The Motes and other Works in proportion accordingly and the Rampires as high as those of the City or Town SECT VII Of some Maxims or Rules necessary to be known in Irregular Fortification IRregular Fortifications is when any Town or Place is to be fortified which lieth in an Irregular form i. e. whose Sides and Angles are unequal in the fortifying of Irregular Figures I shall here say very little only I shall lay down some Precepts that are of immediate concern in fortifying of Irregular Figures and shall refer you to peruse Marlois Dogen Fritach Taurnier Dilichius c. which will greatly satisfie and help you To this end know 1. That the same Laws and Maxims for Regular Fortifications stand and be in force for Irregular i. e. that the line of Defence must not exceed the Port of a Musquet nor the Angles of the Bastion be less than 60° nor much above 90° c. 2. That no inward Angle of the Place be less than 90° if it be so it must be altered and that point may be made the outward point of a Bastion 3. That between Regular and Irregular Fortifications there is no other difference but by rectifying the sides that are too short or too long and altering the Angles that are too little as for the sides if they be above 500 and under 1000 Feet they may be fortified by Bastions placed according to the usual manner at the extreme points thereof But if the sides be between 1000 and 1700 Feet then in the midst you may place a Plat Bastion and at the Extreme Points place two Bastions as before But if the line be less than 500 Feet you may lengthen it by producing it into the Plane As for the Angles they are made greater or lesser according as occasion requireth For the Raising the Rampires Parapets and other Out-works they are to be as in the Regular and the Out-work may be placed before the Curtains as was before mentioned 4. That the Capital in any Regular or Irregular Bastion is found by dividing the Angle of the Polygon into two equal parts by prop. 7. § 1. chap. 4. and by producing the line of Angular Division or Separation on which the due length of the Capital must be placed which observe for a general Rule SECT VIII Of the Dimensions and Construction of small Forts or Scones which are built for the Defence of some Pass River or other place WHEN they are made Regular of 4 5 or 6 Bastions then they may be fortified by the precedent Rules but there are others of smaller Dimensions fit for the same purpose viz. Triangle with Demi-Bastions Square with Demi-Bastions Parallelograms with Demi-Bastions and Tong Star Redoubts of four five or six points and Plain Redoubts PROP. IX To fortifie a Triangle with Demi-Bastions This Triangle may consist and be comprehended of three equal or unequal sides in this Example let it be