A line is said to increase equally, when the poynt describing the same,
goeth forward equall spaces, in equall times, or moments.  I. Definition. 

A to C. In the second mement form C to D.
In the third moment from D to E, & so forth
infinitely, describing the line ACDEF, &c.
The spaces AC, CD, DE, EF, &c. And all
the rest being equall, and described in equall
moments (or times.) This line by the former
definition shall be said to increase equally.  
A Corollary or consequent. 
So also of those quantities AC, and AE, and of these, AH, and AK, the differences CE, and HK are equall, and therefore differing equally, as before. 
2. Definition. 

For examples sake. Let the line of the whole sine aZ be to bee diminished proportionally: let the poynt diminishing the same by his 
motion be b: and let the proportion of each
part to the line from w^{ch} it is cut off, be as QR
to QS. Therefore in what proportion QS is cut in R,
in the same proportion (by the 10 of the 6 of
Euclide) Let aZ be cut in c. and so let
b running from a to c in the first
moment, cut off ac from aZ, the line or sine
cZ remaining.  
 A Corolary. 
by the 19 Prop. 5 and 11. Prop. 7, Euclid.  
3. Def. 

4. Def. 

5. Def. 

6. Def. 

equally in the meane time, whiles the line of the whole sine decreased proportionally into that sine, both motions being equaltimed, and the beginning equally swift. 
As for example. Let the 2 figures going afore bee here repeated, and let B bee moued alwayes, and euery where with equall, or the same swiftnesse wherewith b beganne to bee moued in the beginning, when it was in a. Then in the first moment let B proceed from A to C, and in the same time let b moue proportionally from a to c, the number defining or expressing AC shal be the Logarithme of the line, or sine cZ. Then in the second moment let B bee moued forward from C to D. And in the same moment or time let b be moued proportionally from c to d, the number defining AD, shall be the Logarithme of the sine dZ. So in the third moment let B go forward equally from D to E, and in the same moment let b be moued forward proportionally from d to e, the number expressing AE the Logarithme of the sine eZ. Also in the fourth moment, let B proceed 
to F, and b to f, the number AF shall be the Logarithme of the sine fz. And keeping the same order continually (according to the former definition) the number of AG shall be the Logarithme of the sine gz. AH the Logarithme of the sine hz. AI the Logarithme of the sine iz. AK the Logarithme of the sine kz, and so forth infinitely,  
A cosequet. 


The Logarithmes of Proportionall numbers and quantities are equally differing.  Propos. I. 
 
 Propos. 2. 
 
Propos 3. 

Propos. 4. 

Propos. 5. 

third made lesse by the first, were equall to the fourth: to both sides of the equality adde the first, and the second more by the third shall bee made equall to the fourth, more by the first, which was to be demonstrated.  
 Propos. 6. 
An Admonition.  
Hitherto we haue shewed the making and symptomes of Logarithmes; Now by what kinde of account or method of calculating they may be had, it should here bee shewed. But because we do here set down the whole Tables, and all his Logarithmes with their Sines to euery minute of the quadrant: therfore passinf ouer the doctrine of making Logarithmes, til a fitter time, we make haste to the vse of them: that the vse and profit of the 
thing being first conceiued, the rest may
please the more, being set forth hereafter, or
else displease the lesse, being buried in
silence. For I expect the iudgement and censure
of learned men hereupon, before the
rest rashly published, be exposed to the detraction
of the enuious. 
1 Section. 

2 Section. 

3 Section. 

4 

5 

6 

7 

8 

 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 
21 

22 

23 

24 

25 

26 

27 

 
28 

29 

30 

 
 31 
The Sines, Tangents and Secants
being precisely found in their
Tables, to finde their Logarithmes
as precisely.  Sect. 1. 


the Logarithme of the number giuen 137 remembring, notwithstanding, that the 4 last figures are to be cut off, or for memory sake to be noted thus expresly 315033 0000. Likewise if by the Tangents aboue expressed, 1370505 you shall seeke the Logarithme of the number 137 by the arch of that Tangent 53 degr. 53 min. shall be found by the 25 Section in the middle columne 315179, the Logarithme of that Tangent 1370505 which because it exceedeth 137 the number giuen by foure places, or figures. Therefore 315179 0000 shall be the Logarithme of the number giuen 137; yet this Logarithme is so much lesse exact by how much 1370505 is more vnlike to the number 1370000, or the 10000 fold of the number giuen. But this error exceedeth not ^{505}/_{10000}. Lastly, if you shall seeke the Logarithme of the number giuen 137 by the Sine aboue written 137156 that shall bee found to bee 1986633 000 by this & the 11 Section of the third chapter. In like manner you shall work by the sign + when the number of the figures of the quantitie giuen, exceedeth the number of the figures of the sine that is likest thereto, which seldome happeneth. As if the Logarithme of the number (or discreet quantitie) 232702 bee sought for, you shall finde in the Table, the sine 2327 most like thereto; but it wanteth two figurs. Therefore to the Logarithme hereof, found in the Table (by the 11 Sect. chap. 3) which is, 6063128 let be added two cyphers, the signe + being put betweene, and it shall be made 6063128 + 00 for the Logarithme of the number 232702 which was sought for. 
Therfor as in the first Section going afore, simple and pure Logarithmes are giuen: so in this Section next going before by putting cyphers to them, they become impure. As of the Addition of 210 to 332 is produced +122. Also of the addition of 210 to 192, comes forth 18. 

last figure, as 0 signifieth, and it shall be
made 99609. And the numeral value of that
Logarithme 2306500 (by the 12 and 13 Sections
following of this chapter) is also 99609
the same that was before. 
decimal numbers, such as 100,000,000, 200,000,000,
300,000,000, &c, which are easie to bee
added or abated to or from any other number. 
whether they be pure or impure. 
value is 18871, because the Secant of the
complement of 32 degrees, that is of 58 degrees,
is 1887080, whose two last figures
next the right hand 80, are to be blotted out
for 00 adicyned to the Logarithme. 
which the Logarithmes are made or resolued,
which is that you descend from the sine
giuen by meanes Geometrically proportionall,
vntill you come to the next lesse sine in
the Table. Likewise from the Logarithme
heereof, in the Table, that you descend
also by as many agreeable meanes Arithmeticall:
and the last of these shall be the Logarithme
of the first of them, and contrariwise
by resolution that you descend from the Logarithme
giuen by Arithmetical meanes to
the next lesse Logarithme in the Table, and
from the value of this in the Table likewise,
that you descend, by as many meanes Geometricall
and agreeable: and the last of these
shall bee the numerall value of the first of
those Logarithmes. But what Arithmeticall
equalitie of difference agreeth and is fitting
to euery continued Geometricall proportion,
is a matter of no meane skil to finde out.
Wherefore of these (if God will) we shall intreate
hereafter more at large, when we shall
handle the making of Logarithmes. Of the most ample use of the Logarithmes, and ready practise by them. Of the Logarithmes of three proportionals, the middle Logarithme being giuen, and one extreame, to finde the other extreame, or his proprtionall, or arch by one doubling, or subtraction onely. 
the double of the middle Logarithme giuen,
subtract the giuen Logarithme of the extreame,
and there shall remaine the Logarithme
of the extreame that was sought for:
which being found in the third, fourth, or
fifth columne of the Table, you haue the arch
answering thereto in the first and seuenth columne,
and the Sine in the second or fixth,
and their Secants or Tangents in their Tables,
by the third Chapter, Section 1, 2, 6, 8,
11, 14, 22, 25, 28, 30 for the extreame that
was sought for. 
sine whereof 555570 is the third proportionall
number that was sought for. Let the extremes 1000000 and 500000 bee giuen, and let the meane proportionall be sought: that commonly is found by multiplying the extreames giuen, one by another, and extracting the square root of the product. But we finde is easilier thus; We adde the Logarithmes of the extreames 0 and 693147, the summe whereof is 693147 which we divide by 2 & the quotient 346573 shall be the Logar. of the middle proportionall desired. By which the middle proportionall 707107, and his arch 45 degrees are found as before. Also let the extremes giuen bee 1056256 and 555570, their Logarithmes are 54730 and 587760. The summe of these put together, is 533030 by the 5 Sect. Chap. 4 which we diuide by two, and the quotient is 266515, the Logarithme and his arch 50 degr. and the sine or meane proportionall sought for is 766044 found by addition onely, and diuision by two. 

degrees, and 30 degrees, subtract the Logarithme
of 50 degrees, there shall remaine the
Logarithme of 40 degrees: and so the erch
it selfe 40 degrees, shall be knowne without
the sines, or their multiplication or diuision,
according as we promised in the beginning. 
of the middle numbers, or any of their arches,
onely by diuiding by three, in stead of the troublesome
extracting of the cubicke root. 
and 500000: the first of these being found
in the Table of Secants, the Logarithme
thereof in our Table is 346573, and the
Logarithme of 500000 is 693147 to the
double whereof, 1386294 adde 346573,
the summe shall be 1039720, which diuided
by 3, will be 346573 the Logarithme of the
meane proportional next the lesse extreme,
which is 707107. So to the double of
346573, which is 693147,
add 693147, and there shall be made thereof
nothing, which divided by 3, maketh also 0,
the sine anf the value whereof is 1000000
for the remaining and greater meane proportionall.
These foure therefore are continually
proportionall, 1414213, 1000000,
707107, 500000. 