# Who gave mathematical expression for torque?

I am trying to understand the conceptual development of the "torque". I saw that Archimedes was the one from whom this idea took birth. I wanted to know from whom the torque got its mathematical form.

I will also be happy to know further reading materials and sources.

• By "mathematical expression" do you mean its current form, using the cross product of vectors resulting in a vector, or would a statement of the magnitude of the resulting torque suffice? – Rory Daulton Dec 20 '15 at 12:29
• @RoryDaulton: I wanted to understand torque concept development in "slow motion". So, I am really interested in both the forms from its prototype without vector cloth to its modern form of vectors. Thank you very much for the helping hand. I hope to see you further... – Immortal Player Dec 20 '15 at 12:53

Regarding the word torque, Wiki refers to : James Thomson, (Joseph Larmor editor, 1912), Collected Papers in Physics and Engineering, page xlvii (dated 1888).

Vector analysis was quite slow to be used in physics at the end of 19th Century [see this post for references].

For some 19th Century textbooks, and the "verbal" definition, see:

• James Clerk Maxwell, Matter and motion (1st ed, 1877) : §70. Moment of a force about a point:

the product of a force into the perpendicular from the origin on its line of action is called the Moment of the force about the origin.

and :

§46. The Moment of a velocity or of a force about any point is the product of its magnitude into the perpendicular from the point upon its direction.

The concept of torque, also called moment of a force, as you've pointed out correctly, originated with the studies of Archimedes on levers. The term torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884, as suggested here.

The Mathematical form of Torque as we know it now is $\vec{\tau} = \vec{r} \times \vec{F}$. Or $\tau = r \cdot F \cdot \sin{\theta}$

While studying/using levers the Law Of The Lever was used which just gave us the mechanical advantage of the lever.

Take this scenario: If a and b are distances from the fulcrum to points A and B and the force FA applied to A is the input and the force FB applied at B is the output, the ratio of the velocities of points A and B is given by $\frac{a}{b}$, so we have the ratio of the output force to the input force, or mechanical advantage, is given by $\frac{F_B}{F_A} = \frac{a}{b}.$ Archimedes proved it using Geometric Reasoning, as stated here.

After which Newton's Law's helped grasp rotational dynamics better.

• Thank you for the help. Are you saying Archimedes to have given "math expression" for Torque? – Immortal Player Dec 20 '15 at 6:22
• @Vinaykumar I wouldn't say that, but I guess it helped us in getting there. – Wave Metric Dec 20 '15 at 6:56
• But...my actual question is who gave "mathematical expression" for torque? Of course your material did help, but the question is different. I hope you are understanding me. – Immortal Player Dec 20 '15 at 9:48

Archimedes did provide a mathematical expression for torque in his Law of Levers - but it could be argued that the Peripatetic School beat him to it in the book Mechanica (source)

Archimedes wrote:

Magnitudes are in equilibrium at distances reciprocally proportional to their weights.

ΤΑ ΜΕΓΕΘΕΑ ΙΣΟΡΡΟΠΕΟΝΤΙ ΑΠΟ ΜΑΚΕΩΝ ΑΝΤΙΠΕΠΟΝΘΟΤΩΣ ΤΟΝ ΑΥΤΟΝ ΛΟΓΟΝ ΕΧΟΝΤΩΝ ΤΟΙΣ ΒΑΡΕΣΙΝ.