1

–

2

1

–

2

√s(s– a)(s– b)(s– c)

Hilbert, David 249

rate) study of pressure in fluids, along with a descrip-

tion of a collection of trick gadgets and toys illustrating

specific scientific principles. He also describes designs

for over 100 practical machines, including pneumatic

pulleys and lifts, wind organs, coin-operated machines,

fire engines, and steam-powered engines that operate in

a way similar to today’s jet engine.

Some of Heron’s texts have the appearance of draft

lecture notes, leading some historians to suspect that he

may have taught at the famous Museum of Alexandria.

Little is actually known of Heron’s life.

Heron’s formula (Hero’s formula) In his work Met-

rica, H

ERON OF

A

LEXANDRIA

(ca. 100

C

.

E

.) presented a

formula for the

AREA

of a

TRIANGLE

solely in terms of

the side-lengths of the triangle. Today known as

Heron’s formula, it reads:

area =

where a, b, and care the sides of the triangle, and

s=(a+ b+ c) is its “semiperimeter.” The formula is a

special case of B

RAHMAGUPTA

’

S FORMULA

, discovered

500 years later.

Heron’s formula can be proved as follows: if θis the

angle between the sides of length aand b, then the area

of the triangle is given by area = ab sin(θ). The

LAW

OF COSINES

asserts that c2= a2+ b2–2ab cos(θ). Solving

for sin(θ) and cos(θ), and substituting into the standard

identity from

TRIGONOMETRY

, cos2(θ) + sin2(θ) = 1,

yields, after some algebraic work, Heron’s result.

See also B

RETSCHNEIDER

’

S FORMULA

;

MEDIAN OF A

TRIANGLE

;

QUADRILATERAL

;

TRIANGLE

.

Heron’s method (Hero’s method) In Book I of his

volume Metrica, H

ERON OF

A

LEXANDRIA

gives a

method for approximating the

SQUARE ROOT

of a num-

ber. It works as follows:

Estimate the value of the square root. Divide

this guess into the number under considera-

tion, and take the average of the result and the

initial estimate. This will produce a better

approximation to the square root.

Repeated application of this method pro-

duces an estimate to any desired degree of

accuracy.

In symbols, Heron claims that if xapproximates

the square root of a number N, then is a better

approximation. For example, taking 3 as an approxima-

tion to the square root of 10, we obtain

as an improved estimate. Repeating the procedure yields

as an even better approxima-

tion. (In fact, √

–

10 ≈3.16227766.)

To show why this method works, let .

Then , which shows that if the

error x– √

–

Nis small, then the error y– √

–

Nwill be even

smaller. (We are assuming here that xis a value greater

than 1.)

This method was known to the Babylonians of

2000

B

.

C

.

E

. It is also equivalent to N

EWTON

’

S METHOD

when applied to the function f(x) = x2– N.

See also B

ABYLONIAN MATHEMATICS

.

higher derivative Taking the

DERIVATIVE

of the same

function more than once, if permissible, produces the

higher derivatives of that function. The first, second

and third derivatives of a function f(x) are denoted,

respectively, f′(x),f′′(x) and f′′′(x),and for n≥4, the

nth derivative as f(n)(x). For example, the third deriva-

tive of f(x) = x4+ sin xis f′′′(x) = 24x– cos x. This

notation for the repeated derivative is due to J

OSEPH

-

L

OUIS

L

AGRANGE

(1736–1813). G

OTTFRIED

W

ILHELM

L

EIBNIZ

(1646–1716), coinventor of

CALCULUS

, used the

notation for the higher derivatives, and French

mathematician Louis Arbogast (1759–1803) wrote

Dnf(x). All three notational systems are used today.

Hilbert, David (1862–1943) German Formal logic,

Geometry, Mathematical physics, Algebraic number

theory Born on January 23, 1862, in Königsberg,

Prussia (now Kaliningrad, Russia), mathematician

David Hilbert is remembered as one of the founding

fathers of 20th-century mathematics. In 1899 Hilbert

dnf(x)

–––

–

dxn