NEWTONIAN PHILOSOPHY. 
Lem. 1. Quantities, and the ratios of quan- 
tities, which m any finite time converge con- 
tinually to equality, and before the end of 
that time approach nearer the one to the 
other than by any given difference, become 
ultimately equal. 
Lem. 2. shews, that in a space bound- 
ed by two right lines and a curve, if an infi- 
nite number of parallelograms are inscribed, 
all of equal breadth; then the ultimate ratio 
of the curve space, and the sum of the pa- 
rallelograms, will be a ratio of equality. 
Lem. 3. shews, that the same tiling is true 
when the breadths of the parallelograms are 
unequal. 
In the succeeding lemmas it is shewn, in 
like manner, that tne ultimate ratios of the 
sine, chord, and tangent, of arcs infinitely 
diminished, are ratios of equality; and there- 
fore that in all our reasonings about these, 
we may safely use the one lor the other : that 
the ultimate form of evanscent triangles, 
made by the arc, chord, or tangent, is that 
of similitude, and their ultimate ratio is that 
of equality ; and hence, in reasonings about 
ultimate ratios, these triangles may safely be 
used one for another, whether they are made 
with the sine, the arc, or the tangent. He 
then demonstrates some properties of the 
ordinates of curvilinear figures; and shews 
that the spaces which a body describes by 
any finite force urging it, whether that force 
is determined and immutable, or continually 
varied, are to each other, jn the very begin- 
ning of the motion, in the duplicate ratio of 
the forces; and lastly, having added some 
demonstrations concerning the evanescence 
of angles of contact, he proceeds to lay down 
the mathematical part of his system," which 
depends on the following theorems. 
I heor. 1. i iie areas which revolving bo- 
dies describe by radii drawn to an immoveable 
centre of force, lie in the same immoveable 
planes, and are proportional to the times in 
which they are described. To this proposi- 
tion are annexed several corollaries, respect- 
ing the velocities ot bodies revolving by cen- 
tripetal forces, the directions and proportions 
of those forces, &c. such as, that the velocity 
of such a revolving body is reciprocally as the 
perpendicular let fall from the centre of force 
upon the line touching the orbit in the place 
■of the body, &c. 
I heor. 2. Everybody that moves in any 
curve line described in a plane, and by a 
radius drawn to a point eitaer immoveable 
or moving forward with an uniform rectili- 
near motion, describes about that point areas 
proportional to the times, is urged by a cen- 
tripetal force directed to that point. With 
corollaries relating to such motions in resist- 
ing mediums, and to the direction of the 
forces when the areas are not proportional to 
the times. 
1 heor. 3. Every body that, bv a radius 
drawn to the centre of another bod v, any how 
moved, describes areas, about that centre 
proportional to the times, is urged by a force 
compounded of the centripetal forces tending 
to tint other body, and of the whole accele- 
rative force by which that other body is im- 
pelled. With several corollaries. 
Theor. 4. The centripetal forces of bodies 
which by equal motions describe different 
circles, tend to tire centres of the same cir- 
cles; and are one to the other as the squares 
■Of the arcs described in equal times, applied 
to the radii of the circles. With many co- 
rollaries relating to the velocities, times, pe- 
riodic forces, &c. And, in a scholium, the 
author farther adds, moreover, by means of 
the foregoing proposition and its corollaries, 
we may discover the proportion of a centri- 
petal force to any other known force, such as 
that of gravity. For if a body, by means of 
its gravity, revolves in a circle concentric to 
the earth, this gravity is the centripetal force 
of that body. But from the descent of heavy 
bodies, the time of one entire revolution, as 
well as the arc described in any given time, 
is given by a corollary to this proposition. 
On these and such-like principles depends 
the Newtonian mathematical philosophy. 
1 he author farther shews how to find the 
centre to which the forces impelling any body 
are directed, having the velocity of the body 
given; and finds that the centrifugal force is 
always as the versed sine of the nascent arc 
directly, and as the square of the time in- 
versely; or directly as the square of the velo- 
city, and inversely as the chord ot the nascent 
arc. From these premises, he deduces the 
method of finding the centripetal force di- 
rected to any given point when the body re- 
volves in a circle; and this, whether the cen- 
tral point is near hand, or at immense dis- 
tance; so that all the lines drawn from it may 
be taken for parallels. And lie shews the 
same thing with regard to bodies revolving in 
spirals, ellipses, hyperbolas, or parabolas. 
He shews also, having the figures oi the or- 
bits given, how to find the velocities and 
moving powers ; and indeed resolves the most 
difficult problems relating to the celestial 
bodies with a surprising degree of mathema- 
tical skill. These problems and demonstra- 
tions are all contained in the first book of the 
Principia ; but an account of them here 
would neither be generally understood, nor 
easily comprised in the limits of this work. 
In tiie second book, Newton treats of the 
properties and motion of fluids, and their 
powers of resistance, with the motion of bo- 
dies through such resisting mediums, those 
resistances being. in the ratio of any powers of 
the velocities ; and the motions being either 
made in right lines or curves, or vibrating 
like pendulums. 
On entering upon the third book of the 
Principia, Newton briefly recapitulates the 
contents of the two former books in these 
words: “ In the preceding books I have laid 
down the principles of philosophy, principles 
not philosophical, but mathematical ; such, 
to wit, as we may build our reasonings upon 
in philosophical enquiries. These principles 
are, the laws and conditions of certain mo- 
tions, and powers or forces, which chiefly 
have resp. ct to philosophy. But lest they 
should have appeased of themselves dry and 
barren, T have illustrated them here and there 
with some, philosophical scholiums, giving an 
account of such things as are of a more ge- 
neral nature, and which philosophy seems 
chiefly to be founded on ; such as the density 
and the resistance of bodies, spaces void of 
all matter, and the motion of light and sounds, 
ft remains, he adds, that from the same prin- 
ciples I now demonstrate the frame of the 
system of the world. Upon this subject L 
had indeed composed the third book in a 
popular method, that it might be read by 
many. But afterwards considering that such 
as had not sufficiently entered into the prin- 
271 ' 
ciples could not easily discern the strength of 
the consequences, nor lay aside the preju- 
dices to which they had been many yeart 
accustomed ; therefore to prevent the disputes 
which might be raised upon such accounts, 
I chose to reduce the substance of that book 
into the form of propositions, in the mathe- 
matical way, which should be read by those 
only who had first made themselves masters 
of the principles established in the preceding; 
books.’’ 
As a necessary preliminary to this third 
part, Newton lays down rules for reasoning 
in natural philosophy. 
The phenomena first considered are, 1 . 
That the satellites of Jupiter, by radii drawn 
to his centre, describe areas proportional to 
the times of description ; and that their peri- 
odic times, the fixed stars being at rest, are 
in the sesquiplicate ratio of their distances 
from that centre. 2. The same thing is like- 
wise observed of the phenomena of Saturn. 
3. The five primary planets, Mercury, Ve- 
nus, Mars, Jupiter, Saturn, with their several 
orbits, encompass the sun. 4. The fixed 
stars being supposed at re^ the periodic 
times of the said five primary planets, and of 
the earth, about the sun, are in the sesquipli- 
cate proportion of their mean distances from 
the sun. 5. The primary planets, by radii 
drawn to the earth, describe areas no ways 
proportional to the times; but the areas which 
they describe by radii drawn to the sun are 
proportional to the times of description. 6. 
I he moon, by a radius drawn to the centre 
of the earth, describes an area proportional 
to the time of description. All which phe- 
nomena are clearly evinced by astronomical 
observations. The mathematical demonstra- 
tions are next applied by Newton in -the fol- 
lowing propositions. 
Prop. 1. The forces by which the satellites 
of Jupiter are continually drawn off from rec- 
tilinear motions, and retained in their proper 
orbits, tend to the centre of that planet, and 
are reciprocally as the squares of the distances 
of those satellites from that centre. 
Prop. 2. 1 he same thing is true of the pri- 
mary planets, with respect to the sun’s centre. 
Prop. 3. The same thing is also true of the 
moon, in respect of the earth’s centre. 
Prop. 4. The moon gravitates towards the 
earth ; and b\ the force of gravity is conti- 
nually drawn off from a rectilinear motion, 
and retained in her orbit. 
Prop. 5. The same tiling is true of all the 
other planets, both primary and secondary, 
each with respect to the centre of its motion. 
Prop. 6. All bodies gravitate towards every 
planet; and the weights of bodies towards anv 
one and the same planet, at equal distances 
from its centre, are proportional to the quan- 
tities of matter they contain. 
Prop. 7. There is a power of gravity tend- 
ing to ail bodies, proportional to the several 
quantities of matter which they contain. 
Prop. 3. In two spheres mutually gravi- 
tating each towards the other, if the matter in 
places on all skies, round about and equidis- 
tant from the centres, is similar, the weight 
of either sphere towards the other, will be re- 
ciprocally as the square of the distance be- 
tween their centres. Hence are compared to- 
gether the weights of bodies towards different 
planets; hence also are discovered the quanti- 
ties of matter in the several planets; aud hence 
