Briggs, Henry 53

1

figure: or . (Since the interior

angles of a quadrilateral sum to 360 degrees, each

quantity yields the same value for the cosine.)

Many texts in mathematics state Bretschneider’s

result without proof. Although algebraically detailed,

the derivation of the result is relatively straightforward.

One begins by noting that area Kis the sum of the

areas of triangles ABC and ADC. We have:

Multiplying by four and squaring yields:

16K2= 4a2b2sin2B+ 4c2d2sin2D+ 8abcd sin B sin D

Call this equation (i). Applying the

LAW OF COSINES

to

each of the two triangles yields the relationship:

a2+ b2– 2ab cos B= c2+ d2– 2cd cos D

which can be rewritten:

a2+ b2– c2– d2= 2ab cos B– 2cd cos D

Call this equation (ii). Also note that, with the aid of

this second equation:

4(s– c)(s– d) = (a+ b– c+ d)(a+ b+ c– d)

= a2+ b2– c2– d2+ 2ab + 2cd

= 2ab cos B– 2cd cos D+ 2ab + 2cd

= 2ab(1 + cos B) + 2cd(1 – cos D)

Similarly:

4(s– a)(s– b) = (–a+ b+ c+ d)(a– b+ c+ d)

= 2ab(1–cos B) + 2cd(1 + cos D)

Multiplying these two equations together gives:

16(s– a)(s– b)(s– c)(s– d)

= 4a2b2(1– cos2B) + 4c2d2(1– cos2D)

+ 4abcd((1 + cos B)(1 + cos D)

+ (1–cos B) (1–cos D))

= 4a2b2sin2B+ 4c2d2sin2D+

8abcd(1 + cos Bcos D)

and substituting back into equation (i) produces:

16K2= 16(s– a)(s– b)(s– c)(s– d) –

8abcd (cos Bcos D– sin Bsin D+ 1)

The following identities from trigonometry:

cos Bcos D– sin Bsin D= cos(B+ D)

and

now give:

which directly yields the famous result.

If, further, the opposite angles of the quadrilateral

sum to 180 degrees (in which case the quadrilateral is a

CYCLIC POLYGON

), then Bretschneider’s formula reduces

to B

RAHMAGUPTA

’

S FORMULA

for the area of a cyclic

quadrilateral:

If one of the sides has length zero, say d= 0, then

the quadrilateral is a triangle and we have H

ERON

’

S

FORMULA

for the area of a triangle:

This is valid, since every triangle can be inscribed in a

circle and so is indeed a cyclic polygon.

Briggs, Henry (1561–1630) British Logarithms Born

in February 1561 (the exact birth date is not known) in

Yorkshire, England, scholar Henry Briggs is remem-

bered for the development of base-10 logarithms, revis-

ing the approach first taken by the inventor of

logarithms, J

OHN

N

APIER

(1550–1617). Today such

common logarithms are sometimes called Briggsian log-

arithms. In 1617, after consulting with Napier, Briggs

published logarithmic values of the first 1,000 numbers

and, in 1624, in his famous text The Arithmetic of Log-

arithms, the logarithmic values of another 30,000 num-

bers, all correct to 14 decimal places.

K= √s(s– a)(s– b)(s– c)

K= √(s– a)(s– b)(s– c)(s– d)