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.

SECT. I. Of the Embattelling and Ordering of Soul∣diers.

BATTAILS are considered either in re∣spect 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 Geome∣trical 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 ex∣tract 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 pla∣ced in File, and 184 in Rank, to order that number of Men propounded into a Double Bat∣tail.

[illustration]

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 Qua∣druple. 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 'tis 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, di∣vide 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.