Proposition 13. In acute-angled triangles the square on the side subtending the acute angle is less than the squres on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular toward the acute angle.

The phrase about "twice a rectangle" can be understood to mean two times AC (the side on which the perpendicular falls) times AD (the straight line cut off outside by the perpendicular .) This is the same as our common expression of the law since AD is equal to AB * Cos (BAC). To be more explicit, the area of a rectangle with sides of AC and AD will have an area equal to AC*AB*Cos(BAC).

Euclids proof, in two cases, can be seen at David Joyce's web page on Euclid's Elements. Proposition 12 of book two deals with the obtuse case, and Proposition 13 addresses the proof for acute triangles.

The law of cosines
is best thought of as an extension of the Pythagorean
Theorem, with a term that adjusts if the included angle is not a right
angle. The usual statement of the theorem is descibed in terms of
sides a, b, and c; and opposite angles A, B, and C. The usual expression
is c^{2}=a^{2}+b^{2}-2abCos(C). The theorem
is cyclic about any of the three sides and so it can also be expressed
in the alternate forms a^{2}=b^{2}+c^{2}-2bcCos(A)
and b^{2}=a^{2}+c^{2}-2acCos(B). Since the
cosine of a right angle is zero, each of the equations reduces to
the usual form of the Pythagorean Theorem when the associated angle is
90^{o}.

A common proof of the property in
textbooks today is to draw the angle C at the origin and place B at the
point (a,0) along the x-axis. This leads to the easy declaration that the
coordinates of point A must be at (b*cosC, b*SinC). Then it is easy to
show the proof by applying the distance formula for AB (side c) and squaring
both sides of the expression and some simple trig identities do the rest.

A somewhat prettier proof using only geometry is the proof used by Pitiscus in __ Trigonometriae sive de triangulorum libri quinque__ which is illustrated below. (It was Pitiscus, by the way, who first used the word

(b-a)(b+a)= c(c-2x)

b

and reordering terms gives b

According to Jeff Miller's web site on the First use of some mathematical terms, the application of the name "Law of Cosines" was near the end of the 19th century;

LAW OF COSINES is found in 1895 in Plane and spherical trigonometry, surveying and tables by George Albert Wentworth: "Law of Cosines. ... The square of any side of a triangle is equal to the sum of the squares of the other two sides, diminished by twice their product into the cosine of the included angle."Here is an image of the page.

I have more recently found an 1892 math book, A Treatise on Plane and Spherical Trigonometry: By Edward Albert Bowser which includes both the plane and spherical versions of the Law of Cosines.

The formula, exactly as we might write it today, appears in the trigonometry addendum (pg 305) at the end of John Playfair's 1804 edition of __Elements of Geometry__.

In spherical triangles both sides and angles are usually treated by their angle measure since sides are arc lengths of a great circle. There is a Law of Cosines for the sides and another for the angles. Using capital letters to represent angles, and lower case to represent the opposite sides, the law for sides is given as:

cos a = cos b cos c + sin b sin c cos A .

and the law for angles is given by

cos A = - cos B cos C + Sin B Sin C cos a.

**Earlier Uses:**

Before this first recorded triangular uses, the term seems to have been applied to a property of light. "Law of Cosines. The intensity per unit of area upon a surface of any effect propagated in straight lines is proportional to the cosine of the inclination of the given surface to a plane normal to the direction of propagation." The term is used in this manner as early as 1873 in The Forces of Nature: A Popular Introduction to the Study of Physical Phenomena, By Amédée Guillemin. To distinguish it from the now popular triangle properties, this property of light is now often called Lambert's Cosine law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760. *Wik A google ngram search indicates that this term became popular around the end of the 19th Century, just as the term Cosine Law was being popularized in plane and spherical trigonometry.