Parallelogram of force

The parallelogram of forces is a method for solving (or visualizing) the results of applying two forces to an object.
When more than two forces are involved, the geometry is no longer parallelogrammatic, but the same principles apply. Forces, being vectors are observed to obey the laws of vector addition, and so the overall (resultant) force due to the application of a number of forces can be found geometrically by drawing vector arrows for each force. For example, see Figure 1. This construction has the same result as moving F2 so its tail coincides with the head of F1, and taking the net force as the vector joining the tail of F1 to the head of F2. This procedure can be repeated to add F3 to the resultant F1 + F2, and so forth.
Newton's proof.
Preliminary: the parallelogram of velocity.
Suppose a particle moves at a uniform rate along a line from A to B (Figure 2) in a given time (say, one second), while in the same time, the line AB moves uniformly from its position at AB to a position at DC, remaining parallel to its original orientation throughout. Accounting for both motions, the particle traces the line AC. Because a displacement in a given time is a measure of velocity, the length of AB is a measure of the particle's velocity along AB, the length of AD is a measure of the line's velocity along AD, and the length of AC is a measure of the particle's velocity along AC. The particle's motion is the same as if it had moved with a single velocity along AC.
Newton's proof of the parallelogram of force.
Suppose two forces act on a particle at the origin (the "tails" of the vectors) of Figure 1. Let the lengths of the vectors F1 and F2 represent the velocities the two forces could produce in the particle by acting for a given time, and let the direction of each represent the direction in which they act. Each force acts independently and will produce its particular velocity whether the other force acts or not. At the end of the given time, the particle has "both" velocities. By the above proof, they are equivalent to a single velocity, Fnet. By Newton's second law, this vector is also a measure of the force which would produce that velocity, thus the two forces are equivalent to a single force.
Bernoulli's proof for perpendicular vectors.
We model forces as Euclidean vectors or members of formula_1. Our first assumption is that the resultant of two forces is in fact another force, so that for any two forces formula_2 there is another force formula_3.
Our final assumption is that the resultant of two forces doesn't change when rotated. If formula_4 is any rotation (any orthogonal map for the usual vector space structure of formula_1 with formula_6), then for all forces formula_7
formula_8
Consider two perpendicular forces formula_9 of length formula_10 and formula_11 of length formula_12, with formula_13 being the length of formula_14.
Let formula_15 and formula_16, where formula_17 is the rotation between formula_9 and formula_14, so formula_20. Under the invariance of the rotation, we get
formula_21
Similarly, consider two more forces formula_22 and formula_23. Let formula_24 be the rotation from formula_9 to formula_26: formula_27, which by inspection makes formula_28.
formula_29
Applying these two equations
formula_30
Since formula_31 and formula_32 both lie along formula_14, their lengths are equal formula_34
formula_35
which implies that formula_36 has length formula_37, which is the length of formula_38. Thus for the case where formula_9 and formula_11 are perpendicular, formula_41. However, when combining our two sets of auxiliary forces we used the associativity of formula_42. Using this additional assumption, we will form an additional proof below.
Algebraic proof of the parallelogram of force.
We model forces as Euclidean vectors or members of formula_1. Our first assumption is that the resultant of two forces is in fact another force, so that for any two forces formula_2 there is another force formula_3. We assume commutativity, as these are forces being applied concurrently, so the order shouldn't matter formula_46.
Consider the map
formula_47
If formula_42 is associative, then this map will be linear. Since it also sends formula_49 to formula_49 and formula_51 to formula_51, it must also be the identity map. Thus formula_42 must be equivalent to the normal vector addition operator.
Controversy.
The mathematical proof of the parallelogram of force is not generally accepted to be mathematically valid. Various proofs were developed (chiefly "Duchayla's" and "Poisson's"), and these also caused objections. That the parallelogram of force was true was not questioned, but "why" it was true. Today the parallelogram of force is accepted as an empirical fact, non-reducible to Newton's first principles.