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####
*For the Literary Magazine*.

#### the progress of geometry.

GEOMETRY, which, in its ori-

ginal, was no more than the art of

measuring the earth, has been very

rarely applied to that purpose, in

after times. Its votaries have been

busily engaged in measuring sur-

faces and figures, which can only

exist in the imagination, such as

circles, spheres, cones, and pyra-

mids, of which, whatever applica-

tions have been made to the men-

suration of empyreal spaces, or ce-

lestial bodies, there has seldom been

any practical use made, in ascer-

taining heights and distances upon

the surface of the earth.

Kingdoms, provinces, towns, and

farms have indeed been surveyed,

but geometry has lent but little as-

sistance on those occasions. The

compass and the line have been al-

most the only instruments employed,

and in the use of these the greatest

blunders and inaccuracies are com-

mitted without scruple or compunc-

tion.

We can hardly, indeed, fail of

observing how much slower the

mathematical arts, in general, have

advanced than the mathematical

sciences. Though the former were

the first to start in the progress of

improvement, they appear to have

fallen behind almost from the first.

The rude manner in which Archi-

medes measured the apparent dia-

meter of the sun is well known; and

while that great geometer was in-

vestigating the properties of the

sphere and cylinder with an acute-

ness and depth that have been the

admiration of all succeeding ages, he

was resolving one of the simplest

problems of practical astronomy, in

a more inaccurate manner than

would be suffered in an ordinary sea-

man of modern times. When the

great problem of measuring the cir-

cumference of the earth was first

thought of, the principle upon which

the solution was attempted was

perfectly scientific; but the execu-

tion, though in skilful hands, was

in the highest degree *slovenly* and

inaccurate. The sages of modern

Europe have traversed the globe,

from the equator to the polar circle,

in order to resolve this great prob-

lem, and are still labouring hard, to

give perfect accuracy to their con-

clusions. The academicians of

Greece and Egypt put themselves to

no such inconvenience. One of them,

when he engaged in the inquiry, ne-

ver quitted his observatory; but

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having measured the sun's solstitial

elevation at Alexandria, where he

lived, he took for granted, on report,

that on the same day the sun was in

the zenith of Syene, being seen there

from the bottom of a deep well. He

also maintained, on no better autho-

rity, the distance and bearing of the

two places, and, with such *data,* was

not ashamed to say he had comput-

ed the circumference of the earth.

At a much later period, Norwood

set about determining the circumfe-

rence of the earth, with an accura-

cy as much superior to that of the

Greek geometer as it was inferior

to recent attempts. Having deter-

mined the latitudes of London and

York, by observation, he travelled

from the one place to the other,

measuring along the high road with

a chain, and taking the bearings

with a compass. He was satisfied

with the accuracy of his work:

“When I measured not,” says he,

“*I paced,* and I believe the experi-

ment has come within a *scantling*

of the truth.”

It is instructive to compare these

early essays of practical geometry

with the perfection to which its ope-

rations have now reached, and to con-

sider, that while the artist had made

so little progress, the theorist had

reached many of the sublimest

heights of mathematical speculation;

that the latter has found out the area

of the circle, and calculated its cir-

cumference to more than a hundred

places of decimals, when the former

could hardly divide an arch into mi-

nutes of a degree; and that many

treatises had appeared on the pro-

perties of curve lines, before a

straight one had ever been accu-

rately drawn or measured on the

surface of the earth.

The progress made in the grand

trigonometrical survey of England,

which was begun in 1784, is more

honourable to geometry, than any

practical application of its principles

which has been recorded. In no

long time, we may derive from geo-

metrical skill an exact delineation of

Great Britain, an achievement

which has been so long shamefully

neglected. While the most pro-

found capacities have been stretched

to the utmost in determining the

course, distance, and diameter of a

planet, some hundreds of millions of

miles distant, the real dimensions

of the smallest lake or island on the

surface of our own globe has been

unknown. Men have accurately

surveyed the path which the moon

takes through the aerial spaces,

while they have remained wholly

ignorant of the shortest way between

the two principal towns in their na-

tive country.

The true reason of this difference

in the progress of speculative and

practical geometry lies, perhaps, in

the greater facility with which the

operations of the former are attend-

ed. It may seem at first a little pa-

radoxical to affirm, that it is more

easy to ascertain the diameter of

Saturn, and the days, hours, mi-

nutes, and seconds which it requires

to pass from one point in the heavens

to another, than to determine the

exact distance which separates two

places within sight of each other,

but the truth is, that the instruments

and calculations by which the for-

mer is effected can be managed by

a solitary student in his closet, with

little expence of any thing but of

time, patience, and attention; where-

as much more of these qualities are

demanded to make two *rods* or

chains of the same precise length,

to place them in the same straight

line, and to make the beginning of

one coincide with end of another.