224 

NATURE 
[FEBRUARY 17, 1923 
Definitions and Laws of Motion in the “ Principia.” 
By Sir Grorcr GREENHILL. 
N ACH’S “ Historical Lectures on Mechanics in 
its Development” ought to have a great 
influence on the treatment of the subject, with an 
English version from the Open Court Company. 
Mach has many a shrewd deep criticism to make on 
Newton’s “ Principia,” and the present remarks are 
intended as an amplification of some of his scientific 
animadversion. It would have been worth his while 
to examine the previous state of the theory of dynamics 
to see what laws were current before the statement 
as given by Newton. These laws must have been 
enunciated, not only to give precision to the subject, 
but at the same time to correct and contradict previous 
fallacy and error, and it would be valuable to have 
a record in historical development. 
The First Law must have excited incredulity, as 
contradictory to common observation of a body in 
motion, soon coming to rest of itself; and when the 
heavenly bodies were pointed at, a divine Primum 
Mobile was postulated to keep them going eternally, in 
the pious reflections of Aristotle, quoted in the former 
conventional manner at the end of the “ Principia,” 
and suggesting Napoleon’s criticism of Laplace. 
Axiomata sive Leges Motus. Lex I. Corpus omne 
perseverare in statu suo quiescendi vel movendi 
uniformiter in directum, nisi quatenus a viribus im- 
pressis cogitur statum illum mutare. 
Similar statements can be traced in the writings 
of Aristotle and Plutarch. But it would be more in- 
structive if we were told something of previous ideas 
contradicted in this Law, such as “A body in motion 
will come to rest of itself,” as in observation of daily 
life, ignoring the reason and cause. 
Lex II. Mutationem motus proportionalem esse vi 
motrici impressae, et fieri secundum lineam rectam qua 
vis illa imprimitur, 
A vector change of momentum, motus, is indicated 
here ; but the Law requires amplification in a com- 
mentary-corollary. Motus is quantity of motion, 
called momentum to-day, and mutationem motus 
requires to be qualified as time change, rate of change 
per time change, per unit of time; not per length 
or distance, which would imply energy or vis viva, 
an idea not extant in Newton’s day. 
Translated into our algebraical symbols, quantity 
of matter, guantitas materiae, of Definition I is denoted 
by W, lb. (in French it would be denoted by P, kg., 
for poids, pondus). Here W, the Pondus of our Corpus, 
is measured by weighing it in the scales, corrected for 
buoyancy of the air, and this is an operation susceptible 
to the greatest accuracy in physical measurement. 
The velocity is v, in f/s (feet per second), so that 
the quantity of motion is Wz, lb.-ft. per sec., according 
to Definition II. Velocity v is not so easy to measure 
to equal accuracy as W, ; 
Then, according to Law II, quantity of motion Wy 
acquired (from rest) under a force F acting for ¢ seconds, 
is proportional to Fi (called the impulse), and expressed 
in a proportion, Wy « Ft, leaving the unit of force 
to choice, 
NO. 2781, VOL. 111] 
The absence of the algebraical sign of equality, =, 
will be noticed as not employed in the “ Principia.” 
But in any numerical calculation, equality must be 
introduced by the appropriate constant factor. 
Working with the practical gravitation Unit of 
Force, the only one in use up to fifty years ago, and 
still the only one capable of exact measurement, 
and taking our unit as the gravitation heft of a pound 
weight, the sign of variation, « , is replaced by the 
sign of equality, =, in the variation above by intro- 
ducing g in the right place, and writing it in a homo- 
geneous form ; 
(x)W 2=F 1 
(Ib.) (sec.) (Ib.) (sec.) 
so as to verify when F=W, with v/g=t, v=gt, as in 
a free vertical fall of the body ; t=v/g being the number 
of seconds of descent to acquire velocity v, ft./sec. ; 
in most practical problems it is near enough to take 
g=32, ft./sec., in round numbers. 
Then if s feet is the distance required to get up 
speed v from rest in ¢ seconds, the average velocity 
Set 
Oa 
and multiplying into equation (x) 
2 
G)W =F s 
: (Ib.) (ft.) Ib.) (ft) 
and iS in a free fall, where F=W. 
And in a flying start, from velocity u, 
a8 
alee 
2g 
With these three equations, (1), (2), (3), any two 
of which imply the third, the young engineer may 
carry on for a long time in the linear dynamics, seen 
on the road and railway or in the air, up and down hill, 
getting up speed and checking it again with the breaks. 
After that a variable force # may be intro- 
duced, as in Hooke’s Law of the spring ; a vibratory 
motion investigated, shown off in the pendulum, and 
seen in reciprocating masses of machinery, or a 
carriage body on springs. Here is theory enough to 
keep him going for a year. A 
Then after linear dynamics comes uniplanar dynamics, 
and the notion of rotation is introduced. A familiar 
illustration is always at hand in the door; every 
room has a door. The muscular sense of starting 
and stopping the rotation can be exercised ; also in 
brandishing a stick, poker, bat, or club. ; 
Angular inertia then requires measurement, although 
not mentioned in the “‘ Principia,” not even in “ De 
motu corporum pendulorum” in Book II, or “In 
Horologiis et similibus instrumentis, quae ex rotulis 
commissis constructa sunt,’ where the Corpus may be 
the compound pendulum of a clock oscillating about 
its axle. Then Moment of Inertia requires definition, — 
the scalar sum of the product of every particle of the 
body by the square of its distance from an axis. 

(1)* w=, (2)*2=2(0+u), (3)*W 
