l e m e . l i b r a r y . u t o r o n t o . c a

s t c 6 8 4 8 

v e r . 1 . 0  ( 2 0 1 9 )



An Arithmeticall Militare Treatise, named
STRATIOTICOS:
Compendiously teaching the Science of Numbers,
as well in Fractions as Integers, and so much of the Ru­
les and Æquations Algebraicall and Arte of Numbers 
Cossicall, as are requisite for the Profession of Soldiour.

Together with the Moderne Militare Discipline, Offices, Lawes and
Dueties in euery wel gouerned Campe and Armie to be obserued:
Long since attempted by LEONARD DIGGES Gentleman,
Augmented, digested, and lately finished, by 
THOMAS DIGGES, his Sonne.

Whereto he hath also adioyned certaine Questions of great Ordinaunce,
resolued in his other Treatize of Pyrotechny and great
Artillerie, hereafter to bee published.

VIVET POST FVNERA VIRTVS.

AT LONDON: 
Printed by Henrie Bynneman.
Anno Domini 1579.








A Number is the multitude of Unites sette
togither, as 2. 3. 4. &c.

The Cipra o augmenteth places, but of himselfe signi­
fieth not. To number anye summe nothing else is, but to
declare the value of euerie Figure placed. So many places 
are in your worke, as there be Elementes from the righte
hand to the left. Euery Figure in the first place betokeneth
himselfe, and so towarde the left side, tenne times so muche
as he was the place before.




TO adde is to gather and knit in one many numbers or 
Unites, whereby the whole summe manifestly may ap­
peare, as 3, 15 and 22 maketh 40 the whole &c.




TO subduce or subtray any summe, is wittily to pull a lesse
from a bigger number, or an equall from a like or equall, so that 
that remaine if it be any thing, do appeare.




TO multiplie, is to find of two Numbers a number product
the one in the other augmented, the which so often may
conteyne the number multiplied, as there be of Unites in 
the Multiplicator. The lesse is named the Multiplicator or 




Multiplyer, the other summe, or number to be multiplyed.




TO deuide or parte, is ingeniously to find how oftentimes
the diuisor is conteined in the number to be diuided, or to
part any number into as manye portions as your diuisor
appoynted. Then diuision sheweth onely howe often the
Lesse summe is conteyned in the bigger, as if I woulde di­
uide 40 by 8, that declareth how oftentimes 8 is conteyned
in 40, so find I 5, and thus is my diuision perfourmed. The
lesse summe is the Diuisor or Diuident, the other that number 
to be diuided or diuisible, vnder whome, the two lines or 
paralels must be drawen for your quotient, and the Diuisor 
vnder them.




TO Reduce, is to bring a number of grosse denominati­
on into a smaller or contrarie. Grosse to subtile by Mul­




tiplication, Subtile, to grosse by partition is perfourmed. A
number of grosse Denomination is that, which conteyneth
many other subtiler or smaller, as a Pounde in respecte of 
Shillings. Shillings in comparison to Poundes are a sub­
tile denomination.

Our English mile grounded vpon the Statute, hath or­
deyned three barlie cornes drie and round to make an ynch,
twelue ynches a foote, three foote a yard, fiue yards a pearch, 
& fortie pearches in length, & four of breadth, an acre of land:
so the acre conteyneth 160 pearches, the halfe 80, and the
yarde, whiche is the quarter 40. Twentie yards make a
Score, and fortie pearches a Furlong, 8 furlong a Mile. So
that by Reduction, you may find in the Englishe mile 320
pearches, 1760 yards, 5280 foote, & 63360 ynches, somewhat
greater than the Italian mile of 1000 pace, & 5 foote to a pace.


Of waight according to the Statute of England.


THe English Pennie which is called the Sterling round
without clipping, shal wey 22 graines of wheat dry, & in 
the midst of the eare. 20. pens maketh 1 ounce, 12 ounces maketh 




a pounde, and this weight is called Troy weighte, whiche
Goldsmithes commonly vse. By this whiche is saide, you
may gather the ounce to conteine 440 graynes of wheate,
and the pound 5280. There is another more vsuall weight
called Haberdepoyse, the Scruple conteyneth 20 Graynes of
Barlie, as the Pennie of Troy, the Dramme 3 scruples, the
ounce 8 Drammes, so ye maye perceyue 480 Graynes of 
Barlie in the ounce, and 7680 in a pound: but these rates
often alter.

...

PRogression is a very compendious way to knitte and
gather in one those numbers which do proceede with an 
eauen and equal distance. In all Arithmetical Progression, 
you shall conioyne the extreames, augmenting the number
of those extreames so added, with the number that sheweth 
how many places there be in the whole round of your pro­
gression, the product mediating, as 5. 10. 20. 25. 30. 35. By 
adding the extreames whiche is 5, and 35 encreaseth 40,
multiplie it by the places which is 7, riseth 280, diuided by
2, or mediated, sheweth 140 the whole summe. There is a­
nother kinde of Progression, named Geometricall, whyche 
encreaseth by 2, 3, 4, 5, or mofold infinitely, behold the figure.




TO find the square Radix, or Roote of any number, is to
gather a summe that multiplyed in himselfe, if it be 
quadrate, iustly may make the aforesayde number, or else
it may engender the bigger squares summe in that contey­
ned. A square number is that which resulteth of any num­
ber once multiplyed in himselfe. A quadrate Radix is cal­
led that number, whiche by himselfe multiplyed, yeeldeth a 
square summe. 




To search or pull out the Radix, or roote cubical,  is to get
a summe multiplyed once by himself, and again in the Pro­
duct engendring the number proponed if it be Cubicall. A
number Cubicall is that whiche by multiplication in hym­
selfe, and agayne in that surmounteth, is ingendred. Radix 
or Roote Cubicall, is that number so multiplyed, making
the said summe cubicall. To multiplie anye number cubi­
callye, is to encrease it once in himselfe, and againe that
number with the product, as if I woulde multiplie 4, 4 in
himselfe maketh 16, and foure times 16, is 64.




A Fraction is a Distribution, appointed of a part or parts 
of an Integer. As the Integers take their beginning 
at 1, and continue in number without ende, euen to the 
said Integers, by imagination from one second part, maye
be dissolued, or broken in portions or parts infinite. The 
partes of those simple or principall Fractions, haue also to 
them parts following. To atteyne the knowledge of them, 
acquaynte your selfe with these two termes, Numerator, 
and Denominator. The office and duetie of the firste, is, to 




expresse the number of such parts. The Denominator doth
giue to those parts their names, as 3/4 which is three portions
of a thing diuided in 4. The Numerator is called the Ele­
mentes or Figures that be alway set aboue the short line. 
The Characters or Elements vnder, beare the name of the 
Denominator. 

...

TO reduce, is to bring Integers into Fractions or con­
trarie, yea a fragment of Fractions of diuers Deno­
minations into one, or what name you list.




TO abbreuiate any Fragment, is to bring a Fraction to 
his lest denomination.




TO ad in common Fractions, is to gather and reduce ma­
ny Fragmentes to one summe or principal Fraction.

SVbduction is the taking of the one Fraction from the 
other, the lesse from the greater, or equal from the like




or equall.





Of the Rule of Proportion, com­
monly called the Golden Rule.
Chapter 20.


THis Rule is also called the Rule of three, for as muche
as by three proportionall numbers knownen, it alwaye
searcheth out the fourth proportionall, as if nine yardes of
Veluet cost eleuen poundes, what shall 33 of the same cost.
Heere are three numbers giuen, and a fourth demaunded.
For the resolution of this and all suche like, worke by the 
Rule Ensuing.

The Rule.

Multiplie the last number by the seconde, and diuide the 
Product by the first number, so will your Quotient and re­
mayne discouer the fourth proportionall desired.




AEquation is nothing else but a certain conference of two
numbers being in value Equal, and yet in multitude and 
Denomination different.