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off their coast, where ihey took up their residence, and in

process of time thfey founded Venice. Here navigation and

commerce revived, and from hence they passed successively

to Genoa, the Hans Towns, Portugal, Spain, England, .and

Holland.

Navigation was very imperfect till after the discovery of the

mariner's compass, which was known about the close of the

thirteenth century, after this the navigator launched boldly

into the ocean, having a guide to direct his operations, and

the means of returning to the place from which he set sail.

At this time, however, the kindred sciences, geometry, tri-

gonometry, and astronomy, which now constitute the ground-

work of navigation, were not sufficiently cultivated to afford

the assistance, which they have since been foufid calculated to

impart. The impulse being given, the human genius soon

saw in what way the mathematical scienccfs might be applied

to the art of navigation, this, with the invention of suitable in-

struments, enabled the navigator to proceed, not only to the

most remote places on the globe, but at length to circumna-

vigate the earth.

There are various methods of sailing described in books, of

which the most ancient is denominated plane sailing : this is

defined to be the art of navigating a ship upon principles

deduced from the supposition that the earth is an extended

plane, and is, in fact, no more than the application of Plane

Trigonometry to the solution of the several variations, or

cases, in which the hypothenuse is always the rhomb line,

that the ship sails upon. This method was soon found to be

inaccurate.

The mariner's compass was not applied to the art of naviga-

tion till about the year 1 420, and fifty years afterwards tables

of the sun's declination were calculated for the use of sailprs,

and the astrolabe was then used for taking observations at sea.

After this the use of the cross-stafi" was introduced among

520 MATHEMATICS.

sailors, for observing the distance between the moon and any

given star, in order theuce to determine the longitude. At

this time navigation was very imperfect^ on account of the in<

accuracies of the plane chart, which must have greatly misled

the, mariner, especially in voyages far distant from the equa-

tor. About the year 1545 two Spanish treatises were publish-

ed on the subject, one by Pedro de Medina, the other by

Martin Cortes, the latter contained a complete system of the

art as far as it was then known. About the same time pro-

posals were made for fiuding the longitude by observations of

the moon. Previously to this, in 1530, Gemma Frisius ad-

vised the keeping of time, by means of small clocks or watches,

then, as he says, newly invented. This author contrived a

new sort of cross-staff, and an instrument called the nautical

quadrant. *

In 1537, Pedro Nunez, or Nonius, published a book in

the Portuguese language, in which he exposes the errors of

the plane chart, and gives the solution of many curious astro-

nomical problems ; among which is that for determining the

latitude from two observations of the sun's altitude, an inter-

mediate azimuth being given. In 1577, Mr. William Bourne

published a treatise, in which he advises, in sailing towards

the high latitudes, to keep the reckoning by the globe, as in

those cases the plane chart is most erroneous : he also advises

to keep an account of the observations, as useful for finding

the place of a ship, which advice was prosecuted at large, by

Simon Stevinus, in a treatise, published at Leyden in 1 599,

the substance of which was printed in the same year at Lon.

don, in English, by Mr. Edward Wright, entitled " The

Haven-finding Art." In this tract is described likewise the

way by which our sailors estimate the rate of a ship in her

course, by an instrument called the log, so named from the

piece of wood or log that floats in the water, while the time is

reckoned, during which tlie line that is fastened to it is running

out. The author of this conlj ivauce is not known, nor was it

NAVIGATION. ' 521

noticed till 1607, in an East-Indian voyage, published by

Purchas, and from this time it became famous, and has been

noticed by almost all succeeding writers in navigation.

In 1581, Michael Coignet, a native of Antwerp, published

a treatise, in which he shewed, that as the rhombs are spirals^

making endless revolutions about the poles, numerous errors

must arise from their being represented by straight lines on

sea-charts. Among other things he described the cross-staff,

with three transverse pieces, as it is at present made, which was

then in common use on ship-board. He^likewise gave an ac-

count of some instruments of his own invention, among which

was the nocturnal.

About this period Mr. Robert Norman discovered the dip-

ping needle, and he made considerable improvements in the

construction of compasses themselves. To this work of Nor-

man's is always prefixed a discourse on the variation of the

magnetic needle, by Mr. William Borrough, in which he

shews how to determine the variation in many different

ways.

Globes of an improved kind, of a much larger size than

those formerly used, were constructed, and many improve-

ments were made in other instruments, still the plane chart

was continued to be followed, though its errors were fre-

quently complained of. The methods of removing these was

pointed out by Gerard Mercator, who proposed to represent

the parallels of latitude and longitude by parallel straight

lines, but gradually to augment the number of the former as

they approached the pole. Thus the rhombs, which otherwise

ought to have been curves, were' now extended into straight lines,

and accordingly a straight line drawn between any two places

marked upon the chart, woRld make an angle with the meri-

dians, expressing the rhomb leading from one to the other.

In 1569> Mercator published an universal map, constructed

in this manner, but it does not appear that he was acquainted

with the principles on which it was founded, and iC, seems

' generally agreed, that the true principles, on which the con-

522 ' MATHEMATICS.

struction of what is still denominated Mercator's chart de-

pends, were first discovered by our countryman, Mr. Edward

Wright,* who published, about the year 16(X), his famous

treatise, entitled " The Correction of certain Errors in Na-

vigation," &c. in which he explained very fully the reason of

extending the length of the paiallels of latitude, and the uses

of the chart thus improved, to the purposes of navigation.

In tliis second edition, printed in I6IO, he proposed a method

for determining the nis^gnitude of the earth, and suggested the

idea of making our measures depend upon the length of a

degree on the earth's surface, and not upon the uncertain

lengtli of three barley-corns. He also gave a table of lati-

tudes for dividing the meridian into minutes, having, before

this time, been divided into every tenth minute only. Among

many other improvements, he contrived an amendment in the

tables of the declination and places of the sun and stars, from

his own observations, made with a six-feet instrument in the

years 1594, 5, G, and 7. The improvements of Mr. Wright

soon became known abroad, and a treatise, entitled " Hypom-

nemata Mathematica," was published by Simon Stevinus, for

the use of Prince Maurice, In that part relating to navigation,

the author having treated of sailing on a great circle, and

shewn how to draw the rhombs on a globe, mechanically, inserts

Mr. Wright's tables of latitudes and rhombs, in order to de-

scribe these lines with more accuracy, pretending even to have

discovered an error in Mr. Wright's table, which, however,

the author shewed arose from the slovenly manner of Stevinus's

mode of calculation.

About this period, Lord Napier published an account of

his logarithms, from which Mr. Edmund Gunter constructed

a table of logarithmic sines and* tangents to every minute of

the quadrant, >which he published in 1620. In this work, he

applied to the art of navigation, and other branches of mathe-

matics, his ruler, known by the name of Gimter's scale. He

See RobcrtOQ' Elements of Navigatioa.

Navigation. 523

improved the sector for the same purposes, and shewed how

to take a back observation by the cross-staff ; he described

likewise another instrument of his own invention, called the

eross-bow, for taking the altitudes of the sun and stars.

Gunter's rule was projected into a circular arch by Mr.

Oughtred, in 1633, who explained its uses in a tract, entitled

" The Circles of Proportion." It has since been made in

the form of a sliding ruler.

Napier's tables were first applied to the different cases of

sailing, by Mr. Thomas Addison, in a work entitled " Arith-

metical Navigation," printed in \6'2,5. Mr. Henry Gellibrand

published, in l635, his treatise, entitled " A Discourse ma-

thematical, on the variation of the Magnetical Needle," in

which he pointed out his own discovery of the changes of the

variation.

In the year l635, Mr. Richard Norwood put into execu-

tion the method recommended by Wright, for measuring the

dimensions of the earth, and found a degree on the great

circle of the earth to contain 367,196 English feet, an ac-

count of which he published in his treatise, entitled " The

Seaman's Practice," published in 1637 ; in this tract, he

points out the uses to be made of the fact, in correcting the

errors committed in the division of the log-line ; describes his

own method of setting down, and perfecting a sea-reckoning,

by using a traverse tabl^ ; and how to rectify the course, by

considering the variation of the compass ; and how to discover

currents, and to make a proper allowance on their account.

About the year 1645, Mr. Bond published, in Norwood's

Epitome, an improvement on Wright's method, by a property

in his Meridian line, by which its divisions were more scien-

tifically assigned than by the author himself, which he after-

wards fully explained, in the third edition of Gunter's works,

printed in 1653.

After the true principles of the art had been settled by the^

foregoing writers, the authors on navigation became so nu^^

merous; that it would not at all agree with the limits of our

524 MATHEMATICS.

work, to attempt an enumeration of tlieni. Navigation ha%,

however, been much indebted to Dr. Halley, who perfected

Wright's chart ; to Mr. Henry Briggs, who improved the lo-

garithms invented by Napier ; to Mr. Hadley, for the inven-

tion of the quadrant that bears his name ; and to. the late

Dr. Maskelyne, who was more than forty years Astronomer

Royal, for devising and establishing, under the Commissioners

of Longitude, the Nautical Almanac. Among the later dis-

coveries in this branch of practical science, that of finding the

longitude by lunar observations, and by time-keepers, is the

chief. Dr. Maskelyne put the first in practice, and the time-

keepers constructed by Mr. Harrison, were found to answer

so well, that he obtained the parliamentary reward.

Among the modern authors on navigation, we must mention

Dr. Andrew Mackay's " Theory and Practice of finding the

Longitude at Sea and Land," in two volumes ; this, with his

other works, particularly his " Complete Navigator," and

" Collection of Mathematical Tables," form, it is said, the

most correct and practical system of navigation and nautical

science hitherto published in this country. These, then, with

the " Tables for Navigation and Nautical Astronomy," by

Jos. de Mendoza ; Mr. John Robertson's " Elements of

Navigation," in two vols. 8vo; the " Nautical Almanac,"

and the Tables requisite to be used with it, and the " Biitish

Mariner's Guide," may be considered as a complete library

for a young navigator. Among practical men, Hamilton

Moore's " New Practical Navigator," has long been very

popular, and is still much used. The sixteenth edition was

published in 1804.

MENSURATION.

Mensuration is the art of finding the dimensions and

contents of bodies, by means of others of the same kind ;

thus the length of bodies, or distances, is found by lines, as

yards, feet, inches, &c. ; surfaces by squares, as square inches,

feet, or yards; solids by cubes, as cubic inches, cubic feet,

MENSURATION. S'iS

&c. The invention of this art cannot be traced to any parti-

cular person ; it has usually been given to the Egyptians, by

whom it was probably invented for the purpose of ascertain-

ing the magnitude and relative situation of their lands, after

the waters of the Nile had subsided. Euclid's Elements, it

has been thought, were originally directed to this object; and

many of the beautiful and elegant geometrical propositions in

that work, it is almost certain, arose out of the simple inves-

tigations directed solely to the theory and practical application

of mensuration.

Notwithstanding the perfection to which Euclid attained in

Geometry, the theory of Mensuration was not, in his time,

advanced beyond what related to right-lined figures, which

might be reduced to that of measuring a triangle ; for since all

right-lined figures may be divided into a number of triangles, it

was necessary only to know how to measure these, in order

to find the surface of any other figure whatever, which was

bounded by right lines. After Euclid, Archimedes took up

the theory of mensuration, and carried it to a great extent.

He first found the method of ascertaining the area of a curvi-

linear space, unless the Lunules of Hippocrates are excepted,

which, however, required no other aid than that contained in the

Elements of Euclid. Archimedes found that the area of a

parabola was two-thirds of its circumscribing rectangle. He

also determined the ratio of spheres, spheroids, &c. to their

circumscribing cylinders, and left behind him an attempt at

the quadrature of the circle. He investigated, and determined

to a considerable degree of accuracy, the approximate ratio

between the circumference and diameter of a circle. He

moreover determined the relation between the circle and

ellipse, as well as that of their similar parts, besides which,

he left a treatise on the Spiral. '

Little more of importance was done to advance the science

of mensuration,' till the time of Cavelleri, an Italian mathe-

matician, who flourished in the 'seventeenth century. Before

his time, the regular figures circumscribed about the circle, as

526 MATHEMATICS.

well as those inscribed in it, were always considered as being

limited, both as to the number of sides, and the length of

each. He was the person that introduced the idea of a circle

being a polygon of an infinite number of sides, each of which

was, of course, indefinitely small ; he also considered solids as

made up of an infinite number of sections iiidefmitely thin.

This was the foundation of the doctrine of indivisibles ^ which

was very general iij its application to a variety of difficult pro-

blems, and which was embraced by many eminent mathema-

ticians, such as Huygens, VVallis, and James Gregory. It

was, however, disapproved by other men, celebrated also for

great talents and deep geometrical learning, and particularly

by Sir Isaac Newton, who, among his numerous and brilliant

discoveries, produced his method of fluxions, the excellency

and generality of which, almost instantly superseded that of

indivisibles. Hopes were now revived of squaring the circle,

and the quadrature was attempted with great eagerness ; but,

after many inefl^ectual efforts, it was abandoned ; and mathe-

maticians began to content themselves with finding, by means

of fluxions, ihe most convenient series for approximating to-

wards the true length of this and other curves, and the theory

of mensuration began to make a rapid progress towards per-

fection. Many of the rules were published in the Transac-

tions of Learned Societies, or in separate and detached works,

tiJl, at length, Dr. Hutton' formed them into a complete work,

entitled " A Treatise on Mensuration, in which the several

rules are all demonstrated." Before this time, Hawney's

" Complete Measurer," and a treatise on the subject by Mr.

Robertson, were the only works that could be jeferred to,

either by the artizan or mathematician. Since Dr. Hutton's

publication, which was first given to the world in 4to, and

has since been printed in 8vo, Mr. Bonnycastle has published

an excellent little work on this subject, entitled " An Intro-

duction to Mensuration and Practical Geometry, with Notes,

containing the reason of every rule concisely and clearly de-

inomtrated." The author has very judiciously given jn thp

SURVEYING. 527

text the rules in words at length, with examples to exercise

them; the remarks and demonstrations are confined to the

notes, and may be consulted or not, as shall be thought ne-

cessary ; but, to those who would wish to be acquainted with

the grounds and rationale of the operations which they per-

form, the demonstrations will be found extremely useful ; and

Mr. Bonnycastle has done all in his power to make them

easy. He has, he says, through the whole, " endeavoured to

consult the wants of the learner, more than those of the man

of science," arid hence his work may be strongly recom-

mended to those who would study the subject from the be-

ginning.

SURVEYING.

The art of surveying consists in determining the boundaries

of an extended surface. When applied to the measuring of

land, it comprises the three following parts, viz. taking the

dimensions of the given tract of land ; the delineating or lay-

ing down the same in a map or draught; and finding the su-

perficial content or area of the same. The first of these is

what is properly called surveying ; the second is called plot-

ting or protracting, or mapping ; and the third, casting up or

computing the contents.

Surveying, when performed in the completest manner, says

Mr. Professor Leslie, ^' ascertains the positions of all the

prominent objects within the scope of observation, measures

their mutual distances and relative heights, and consequently

defines the various contours which mark the surface. But

the land-surveyor seldom aims at such minute and scrupulous

accuracy ; his main object is, to trace expeditiously the chief

boundaries, and to compute the superficial contents of each

field. In hilly grounds, however, it is not the absolute sur-

face that is measured, but the diminished quantity that would

result, had the whole been reduced to a horizontal plane. This

distinction is founded on the obvious principle, that, since

plants shoot up vertically, the vegetable produce of a swelling

528 \ MATHEMATICS.

eminence, can never exceed what would liave grown from its

levelled base. All the sloping distances, therefore, are re-

duced invariably to their horizontal lengths, before the calcu-

lation is b^un."

The instruments usually employed in surveying, are the

chain, the plain-table, the cross, and the theodolite. The

English chain is twenty-two yards in length, that is, the tenth

part of a furlong, or the eightieth part of a mile. The chain

is divided into a hundred links, each 7-92 inches in length.

An acre contains ten square chains, or 100,000 links.

When land is surveyed by means of the chain simply, the

several fields are divided into large triangles, of which the

sides are measured by the chain; and if the exterior boundary

happens to be irregular, the perpendicular distance or offset

is taken at each bending. The surface of all the triangles is

then computed by the elements of plane geometry, and the

exterior border of the polygon is considered as a collection of

trapezoids, .which are measured by multiplyuig the mean of

each pair of offsets or perpendiculars, into their base or inter-

mediate distance. In this method, the triangles should be

chosen as nearly equilateral as possible ; for, if they are very

oblique, small errors in the lengths of their sides will occasion

very large ones in the estimate of the surface.

The usual mode of surveying a large estate is, to measure

round it with the chain, and observe the angles at each turn

by means of the theodolite; but the observations must be

taken with great care. If the boundaries of the estate be to-

lerably regular, it may be considered as a polygon, of which

the angles, being necessarily very oblique, are apt, unless

much attention be exercised, to affect the accuracy of

the results. The best method of surveying is, undoubtedly,

to cover the ground with a series of connected triangles,

planting the theodolite at each angular point, and computing

from some base of considerable extent, which has been se-

lected and measured with as much precision as the nature of

the case will admit ^ for angles can be measured more accu-

SURVEYING. 529

mtely than lines ; and hence it has been recommended, that

surveyors should generally employ theodolites of a good con-

struction, and trust as little as possible to the aid of the

chain.

In surveying, for sale or other purposes, large tracts of

land in rude and uncultivated countries, the contents are

usually estimated by the square mile, which includes six hun-

dred and forty acres : thus in the back settlements of North

America, the lands are divided and allotted merely by running

lines north and south, and iuteiseciing iliem by perpendiculars

at each interval of a mile.

We may farther observe, that where any degree of nicety is

required, as is the case in surveying estates of value, the

operator will have frequent occasion for calculation, and

Uierefore it is necessary that he should be familiar with the

four first rules of arithmetic, and the rule of proportion, as

well in fractions and decimals as in whole numbers ; he should

be conversant with the nature and practice of logarithms ;

and if he is acquainted with the elementary parts of algebra, he

will find the advantage of it. As he will have to investigate

and measure lines and angles, and to describe them on paper,

he should well understand and be quick in the application of

the principles of geometry and plane trigonometry.

Dr. Hutton's mensuration will be found to contain an out-

line of the theory and practice of the art of Surveying.

There are several other very respectable treatises on the sub-

ject, by Leadbeater, Wilson, and Stephenson : but the two

works with which we are best acquainted, is one by Mr.

Abraham Crocker, in which will be found several improve-

ments in the art ; and " A Complete Treatise on Land-Survey-

ing by tiie Chain, Cross, and Offset-staffs only. By William

Davis." This treatise is divided into three parts: (1.) It gives

an outline of Practical Geometry, at least such parts of it as

are requisite for Surveying ; and Plane Trigonometry, with its

application to measuring heights and distances. (2,) It goes

through the whole practice of Surveying by the different me-

VOL. I. 2 m

530 MATHEMATICS.

thods ; and (3.) it points out the practical method of obtaining

the contents of Hay-ricks, Pits, Timber, and all kinds of

Artificers' works : likewise the method of levelling, conveying

water from one place to another, and of draining and flooding

land.

Another, but very different branch of this art is denominated

Maritime Surveying, which determines the positions of the

remarkable headlands, and other conspicuous objects that pre-

sent themselves along the coast, or its immediate neighbour-

hood. It likewise ascertains the situaiions of the various inlets,

rocks, shallows, and soundings, which occur in approaching the

shore. The method of performing this, given by Mr. Professor

Leslie, is as follows : " To survey a new or inaccessible coast,

two boats are moored at a proper interval, which is carefully

measured on the surface of the water ; and from each boat

the bearings of all the prominent points of land are taken by

means of an azimuth compass ; or the angles subtended by

these points and the other boat, are measured by a Hadley's

sextant. Having now on paper drawn the base to any scale,

straight lines radiating from each end at the observed angles,

will,^ by their intersections, give the positions of the several

points from which the coast may be sketched. But a chart

is more accurately constructed, by combining a survey made

on land, with observations taken on the water. A smooth

level piece of ground is chosen, on which a base of consider-

able length is measured out, and station staves * are fixed at

its extremities. If no such place can be found, the mutual

distance and position of two points conveniently situate for

planting the staves, though divided by a broken surface, are

process of time thfey founded Venice. Here navigation and

commerce revived, and from hence they passed successively

to Genoa, the Hans Towns, Portugal, Spain, England, .and

Holland.

Navigation was very imperfect till after the discovery of the

mariner's compass, which was known about the close of the

thirteenth century, after this the navigator launched boldly

into the ocean, having a guide to direct his operations, and

the means of returning to the place from which he set sail.

At this time, however, the kindred sciences, geometry, tri-

gonometry, and astronomy, which now constitute the ground-

work of navigation, were not sufficiently cultivated to afford

the assistance, which they have since been foufid calculated to

impart. The impulse being given, the human genius soon

saw in what way the mathematical scienccfs might be applied

to the art of navigation, this, with the invention of suitable in-

struments, enabled the navigator to proceed, not only to the

most remote places on the globe, but at length to circumna-

vigate the earth.

There are various methods of sailing described in books, of

which the most ancient is denominated plane sailing : this is

defined to be the art of navigating a ship upon principles

deduced from the supposition that the earth is an extended

plane, and is, in fact, no more than the application of Plane

Trigonometry to the solution of the several variations, or

cases, in which the hypothenuse is always the rhomb line,

that the ship sails upon. This method was soon found to be

inaccurate.

The mariner's compass was not applied to the art of naviga-

tion till about the year 1 420, and fifty years afterwards tables

of the sun's declination were calculated for the use of sailprs,

and the astrolabe was then used for taking observations at sea.

After this the use of the cross-stafi" was introduced among

520 MATHEMATICS.

sailors, for observing the distance between the moon and any

given star, in order theuce to determine the longitude. At

this time navigation was very imperfect^ on account of the in<

accuracies of the plane chart, which must have greatly misled

the, mariner, especially in voyages far distant from the equa-

tor. About the year 1545 two Spanish treatises were publish-

ed on the subject, one by Pedro de Medina, the other by

Martin Cortes, the latter contained a complete system of the

art as far as it was then known. About the same time pro-

posals were made for fiuding the longitude by observations of

the moon. Previously to this, in 1530, Gemma Frisius ad-

vised the keeping of time, by means of small clocks or watches,

then, as he says, newly invented. This author contrived a

new sort of cross-staff, and an instrument called the nautical

quadrant. *

In 1537, Pedro Nunez, or Nonius, published a book in

the Portuguese language, in which he exposes the errors of

the plane chart, and gives the solution of many curious astro-

nomical problems ; among which is that for determining the

latitude from two observations of the sun's altitude, an inter-

mediate azimuth being given. In 1577, Mr. William Bourne

published a treatise, in which he advises, in sailing towards

the high latitudes, to keep the reckoning by the globe, as in

those cases the plane chart is most erroneous : he also advises

to keep an account of the observations, as useful for finding

the place of a ship, which advice was prosecuted at large, by

Simon Stevinus, in a treatise, published at Leyden in 1 599,

the substance of which was printed in the same year at Lon.

don, in English, by Mr. Edward Wright, entitled " The

Haven-finding Art." In this tract is described likewise the

way by which our sailors estimate the rate of a ship in her

course, by an instrument called the log, so named from the

piece of wood or log that floats in the water, while the time is

reckoned, during which tlie line that is fastened to it is running

out. The author of this conlj ivauce is not known, nor was it

NAVIGATION. ' 521

noticed till 1607, in an East-Indian voyage, published by

Purchas, and from this time it became famous, and has been

noticed by almost all succeeding writers in navigation.

In 1581, Michael Coignet, a native of Antwerp, published

a treatise, in which he shewed, that as the rhombs are spirals^

making endless revolutions about the poles, numerous errors

must arise from their being represented by straight lines on

sea-charts. Among other things he described the cross-staff,

with three transverse pieces, as it is at present made, which was

then in common use on ship-board. He^likewise gave an ac-

count of some instruments of his own invention, among which

was the nocturnal.

About this period Mr. Robert Norman discovered the dip-

ping needle, and he made considerable improvements in the

construction of compasses themselves. To this work of Nor-

man's is always prefixed a discourse on the variation of the

magnetic needle, by Mr. William Borrough, in which he

shews how to determine the variation in many different

ways.

Globes of an improved kind, of a much larger size than

those formerly used, were constructed, and many improve-

ments were made in other instruments, still the plane chart

was continued to be followed, though its errors were fre-

quently complained of. The methods of removing these was

pointed out by Gerard Mercator, who proposed to represent

the parallels of latitude and longitude by parallel straight

lines, but gradually to augment the number of the former as

they approached the pole. Thus the rhombs, which otherwise

ought to have been curves, were' now extended into straight lines,

and accordingly a straight line drawn between any two places

marked upon the chart, woRld make an angle with the meri-

dians, expressing the rhomb leading from one to the other.

In 1569> Mercator published an universal map, constructed

in this manner, but it does not appear that he was acquainted

with the principles on which it was founded, and iC, seems

' generally agreed, that the true principles, on which the con-

522 ' MATHEMATICS.

struction of what is still denominated Mercator's chart de-

pends, were first discovered by our countryman, Mr. Edward

Wright,* who published, about the year 16(X), his famous

treatise, entitled " The Correction of certain Errors in Na-

vigation," &c. in which he explained very fully the reason of

extending the length of the paiallels of latitude, and the uses

of the chart thus improved, to the purposes of navigation.

In tliis second edition, printed in I6IO, he proposed a method

for determining the nis^gnitude of the earth, and suggested the

idea of making our measures depend upon the length of a

degree on the earth's surface, and not upon the uncertain

lengtli of three barley-corns. He also gave a table of lati-

tudes for dividing the meridian into minutes, having, before

this time, been divided into every tenth minute only. Among

many other improvements, he contrived an amendment in the

tables of the declination and places of the sun and stars, from

his own observations, made with a six-feet instrument in the

years 1594, 5, G, and 7. The improvements of Mr. Wright

soon became known abroad, and a treatise, entitled " Hypom-

nemata Mathematica," was published by Simon Stevinus, for

the use of Prince Maurice, In that part relating to navigation,

the author having treated of sailing on a great circle, and

shewn how to draw the rhombs on a globe, mechanically, inserts

Mr. Wright's tables of latitudes and rhombs, in order to de-

scribe these lines with more accuracy, pretending even to have

discovered an error in Mr. Wright's table, which, however,

the author shewed arose from the slovenly manner of Stevinus's

mode of calculation.

About this period, Lord Napier published an account of

his logarithms, from which Mr. Edmund Gunter constructed

a table of logarithmic sines and* tangents to every minute of

the quadrant, >which he published in 1620. In this work, he

applied to the art of navigation, and other branches of mathe-

matics, his ruler, known by the name of Gimter's scale. He

See RobcrtOQ' Elements of Navigatioa.

Navigation. 523

improved the sector for the same purposes, and shewed how

to take a back observation by the cross-staff ; he described

likewise another instrument of his own invention, called the

eross-bow, for taking the altitudes of the sun and stars.

Gunter's rule was projected into a circular arch by Mr.

Oughtred, in 1633, who explained its uses in a tract, entitled

" The Circles of Proportion." It has since been made in

the form of a sliding ruler.

Napier's tables were first applied to the different cases of

sailing, by Mr. Thomas Addison, in a work entitled " Arith-

metical Navigation," printed in \6'2,5. Mr. Henry Gellibrand

published, in l635, his treatise, entitled " A Discourse ma-

thematical, on the variation of the Magnetical Needle," in

which he pointed out his own discovery of the changes of the

variation.

In the year l635, Mr. Richard Norwood put into execu-

tion the method recommended by Wright, for measuring the

dimensions of the earth, and found a degree on the great

circle of the earth to contain 367,196 English feet, an ac-

count of which he published in his treatise, entitled " The

Seaman's Practice," published in 1637 ; in this tract, he

points out the uses to be made of the fact, in correcting the

errors committed in the division of the log-line ; describes his

own method of setting down, and perfecting a sea-reckoning,

by using a traverse tabl^ ; and how to rectify the course, by

considering the variation of the compass ; and how to discover

currents, and to make a proper allowance on their account.

About the year 1645, Mr. Bond published, in Norwood's

Epitome, an improvement on Wright's method, by a property

in his Meridian line, by which its divisions were more scien-

tifically assigned than by the author himself, which he after-

wards fully explained, in the third edition of Gunter's works,

printed in 1653.

After the true principles of the art had been settled by the^

foregoing writers, the authors on navigation became so nu^^

merous; that it would not at all agree with the limits of our

524 MATHEMATICS.

work, to attempt an enumeration of tlieni. Navigation ha%,

however, been much indebted to Dr. Halley, who perfected

Wright's chart ; to Mr. Henry Briggs, who improved the lo-

garithms invented by Napier ; to Mr. Hadley, for the inven-

tion of the quadrant that bears his name ; and to. the late

Dr. Maskelyne, who was more than forty years Astronomer

Royal, for devising and establishing, under the Commissioners

of Longitude, the Nautical Almanac. Among the later dis-

coveries in this branch of practical science, that of finding the

longitude by lunar observations, and by time-keepers, is the

chief. Dr. Maskelyne put the first in practice, and the time-

keepers constructed by Mr. Harrison, were found to answer

so well, that he obtained the parliamentary reward.

Among the modern authors on navigation, we must mention

Dr. Andrew Mackay's " Theory and Practice of finding the

Longitude at Sea and Land," in two volumes ; this, with his

other works, particularly his " Complete Navigator," and

" Collection of Mathematical Tables," form, it is said, the

most correct and practical system of navigation and nautical

science hitherto published in this country. These, then, with

the " Tables for Navigation and Nautical Astronomy," by

Jos. de Mendoza ; Mr. John Robertson's " Elements of

Navigation," in two vols. 8vo; the " Nautical Almanac,"

and the Tables requisite to be used with it, and the " Biitish

Mariner's Guide," may be considered as a complete library

for a young navigator. Among practical men, Hamilton

Moore's " New Practical Navigator," has long been very

popular, and is still much used. The sixteenth edition was

published in 1804.

MENSURATION.

Mensuration is the art of finding the dimensions and

contents of bodies, by means of others of the same kind ;

thus the length of bodies, or distances, is found by lines, as

yards, feet, inches, &c. ; surfaces by squares, as square inches,

feet, or yards; solids by cubes, as cubic inches, cubic feet,

MENSURATION. S'iS

&c. The invention of this art cannot be traced to any parti-

cular person ; it has usually been given to the Egyptians, by

whom it was probably invented for the purpose of ascertain-

ing the magnitude and relative situation of their lands, after

the waters of the Nile had subsided. Euclid's Elements, it

has been thought, were originally directed to this object; and

many of the beautiful and elegant geometrical propositions in

that work, it is almost certain, arose out of the simple inves-

tigations directed solely to the theory and practical application

of mensuration.

Notwithstanding the perfection to which Euclid attained in

Geometry, the theory of Mensuration was not, in his time,

advanced beyond what related to right-lined figures, which

might be reduced to that of measuring a triangle ; for since all

right-lined figures may be divided into a number of triangles, it

was necessary only to know how to measure these, in order

to find the surface of any other figure whatever, which was

bounded by right lines. After Euclid, Archimedes took up

the theory of mensuration, and carried it to a great extent.

He first found the method of ascertaining the area of a curvi-

linear space, unless the Lunules of Hippocrates are excepted,

which, however, required no other aid than that contained in the

Elements of Euclid. Archimedes found that the area of a

parabola was two-thirds of its circumscribing rectangle. He

also determined the ratio of spheres, spheroids, &c. to their

circumscribing cylinders, and left behind him an attempt at

the quadrature of the circle. He investigated, and determined

to a considerable degree of accuracy, the approximate ratio

between the circumference and diameter of a circle. He

moreover determined the relation between the circle and

ellipse, as well as that of their similar parts, besides which,

he left a treatise on the Spiral. '

Little more of importance was done to advance the science

of mensuration,' till the time of Cavelleri, an Italian mathe-

matician, who flourished in the 'seventeenth century. Before

his time, the regular figures circumscribed about the circle, as

526 MATHEMATICS.

well as those inscribed in it, were always considered as being

limited, both as to the number of sides, and the length of

each. He was the person that introduced the idea of a circle

being a polygon of an infinite number of sides, each of which

was, of course, indefinitely small ; he also considered solids as

made up of an infinite number of sections iiidefmitely thin.

This was the foundation of the doctrine of indivisibles ^ which

was very general iij its application to a variety of difficult pro-

blems, and which was embraced by many eminent mathema-

ticians, such as Huygens, VVallis, and James Gregory. It

was, however, disapproved by other men, celebrated also for

great talents and deep geometrical learning, and particularly

by Sir Isaac Newton, who, among his numerous and brilliant

discoveries, produced his method of fluxions, the excellency

and generality of which, almost instantly superseded that of

indivisibles. Hopes were now revived of squaring the circle,

and the quadrature was attempted with great eagerness ; but,

after many inefl^ectual efforts, it was abandoned ; and mathe-

maticians began to content themselves with finding, by means

of fluxions, ihe most convenient series for approximating to-

wards the true length of this and other curves, and the theory

of mensuration began to make a rapid progress towards per-

fection. Many of the rules were published in the Transac-

tions of Learned Societies, or in separate and detached works,

tiJl, at length, Dr. Hutton' formed them into a complete work,

entitled " A Treatise on Mensuration, in which the several

rules are all demonstrated." Before this time, Hawney's

" Complete Measurer," and a treatise on the subject by Mr.

Robertson, were the only works that could be jeferred to,

either by the artizan or mathematician. Since Dr. Hutton's

publication, which was first given to the world in 4to, and

has since been printed in 8vo, Mr. Bonnycastle has published

an excellent little work on this subject, entitled " An Intro-

duction to Mensuration and Practical Geometry, with Notes,

containing the reason of every rule concisely and clearly de-

inomtrated." The author has very judiciously given jn thp

SURVEYING. 527

text the rules in words at length, with examples to exercise

them; the remarks and demonstrations are confined to the

notes, and may be consulted or not, as shall be thought ne-

cessary ; but, to those who would wish to be acquainted with

the grounds and rationale of the operations which they per-

form, the demonstrations will be found extremely useful ; and

Mr. Bonnycastle has done all in his power to make them

easy. He has, he says, through the whole, " endeavoured to

consult the wants of the learner, more than those of the man

of science," arid hence his work may be strongly recom-

mended to those who would study the subject from the be-

ginning.

SURVEYING.

The art of surveying consists in determining the boundaries

of an extended surface. When applied to the measuring of

land, it comprises the three following parts, viz. taking the

dimensions of the given tract of land ; the delineating or lay-

ing down the same in a map or draught; and finding the su-

perficial content or area of the same. The first of these is

what is properly called surveying ; the second is called plot-

ting or protracting, or mapping ; and the third, casting up or

computing the contents.

Surveying, when performed in the completest manner, says

Mr. Professor Leslie, ^' ascertains the positions of all the

prominent objects within the scope of observation, measures

their mutual distances and relative heights, and consequently

defines the various contours which mark the surface. But

the land-surveyor seldom aims at such minute and scrupulous

accuracy ; his main object is, to trace expeditiously the chief

boundaries, and to compute the superficial contents of each

field. In hilly grounds, however, it is not the absolute sur-

face that is measured, but the diminished quantity that would

result, had the whole been reduced to a horizontal plane. This

distinction is founded on the obvious principle, that, since

plants shoot up vertically, the vegetable produce of a swelling

528 \ MATHEMATICS.

eminence, can never exceed what would liave grown from its

levelled base. All the sloping distances, therefore, are re-

duced invariably to their horizontal lengths, before the calcu-

lation is b^un."

The instruments usually employed in surveying, are the

chain, the plain-table, the cross, and the theodolite. The

English chain is twenty-two yards in length, that is, the tenth

part of a furlong, or the eightieth part of a mile. The chain

is divided into a hundred links, each 7-92 inches in length.

An acre contains ten square chains, or 100,000 links.

When land is surveyed by means of the chain simply, the

several fields are divided into large triangles, of which the

sides are measured by the chain; and if the exterior boundary

happens to be irregular, the perpendicular distance or offset

is taken at each bending. The surface of all the triangles is

then computed by the elements of plane geometry, and the

exterior border of the polygon is considered as a collection of

trapezoids, .which are measured by multiplyuig the mean of

each pair of offsets or perpendiculars, into their base or inter-

mediate distance. In this method, the triangles should be

chosen as nearly equilateral as possible ; for, if they are very

oblique, small errors in the lengths of their sides will occasion

very large ones in the estimate of the surface.

The usual mode of surveying a large estate is, to measure

round it with the chain, and observe the angles at each turn

by means of the theodolite; but the observations must be

taken with great care. If the boundaries of the estate be to-

lerably regular, it may be considered as a polygon, of which

the angles, being necessarily very oblique, are apt, unless

much attention be exercised, to affect the accuracy of

the results. The best method of surveying is, undoubtedly,

to cover the ground with a series of connected triangles,

planting the theodolite at each angular point, and computing

from some base of considerable extent, which has been se-

lected and measured with as much precision as the nature of

the case will admit ^ for angles can be measured more accu-

SURVEYING. 529

mtely than lines ; and hence it has been recommended, that

surveyors should generally employ theodolites of a good con-

struction, and trust as little as possible to the aid of the

chain.

In surveying, for sale or other purposes, large tracts of

land in rude and uncultivated countries, the contents are

usually estimated by the square mile, which includes six hun-

dred and forty acres : thus in the back settlements of North

America, the lands are divided and allotted merely by running

lines north and south, and iuteiseciing iliem by perpendiculars

at each interval of a mile.

We may farther observe, that where any degree of nicety is

required, as is the case in surveying estates of value, the

operator will have frequent occasion for calculation, and

Uierefore it is necessary that he should be familiar with the

four first rules of arithmetic, and the rule of proportion, as

well in fractions and decimals as in whole numbers ; he should

be conversant with the nature and practice of logarithms ;

and if he is acquainted with the elementary parts of algebra, he

will find the advantage of it. As he will have to investigate

and measure lines and angles, and to describe them on paper,

he should well understand and be quick in the application of

the principles of geometry and plane trigonometry.

Dr. Hutton's mensuration will be found to contain an out-

line of the theory and practice of the art of Surveying.

There are several other very respectable treatises on the sub-

ject, by Leadbeater, Wilson, and Stephenson : but the two

works with which we are best acquainted, is one by Mr.

Abraham Crocker, in which will be found several improve-

ments in the art ; and " A Complete Treatise on Land-Survey-

ing by tiie Chain, Cross, and Offset-staffs only. By William

Davis." This treatise is divided into three parts: (1.) It gives

an outline of Practical Geometry, at least such parts of it as

are requisite for Surveying ; and Plane Trigonometry, with its

application to measuring heights and distances. (2,) It goes

through the whole practice of Surveying by the different me-

VOL. I. 2 m

530 MATHEMATICS.

thods ; and (3.) it points out the practical method of obtaining

the contents of Hay-ricks, Pits, Timber, and all kinds of

Artificers' works : likewise the method of levelling, conveying

water from one place to another, and of draining and flooding

land.

Another, but very different branch of this art is denominated

Maritime Surveying, which determines the positions of the

remarkable headlands, and other conspicuous objects that pre-

sent themselves along the coast, or its immediate neighbour-

hood. It likewise ascertains the situaiions of the various inlets,

rocks, shallows, and soundings, which occur in approaching the

shore. The method of performing this, given by Mr. Professor

Leslie, is as follows : " To survey a new or inaccessible coast,

two boats are moored at a proper interval, which is carefully

measured on the surface of the water ; and from each boat

the bearings of all the prominent points of land are taken by

means of an azimuth compass ; or the angles subtended by

these points and the other boat, are measured by a Hadley's

sextant. Having now on paper drawn the base to any scale,

straight lines radiating from each end at the observed angles,

will,^ by their intersections, give the positions of the several

points from which the coast may be sketched. But a chart

is more accurately constructed, by combining a survey made

on land, with observations taken on the water. A smooth

level piece of ground is chosen, on which a base of consider-

able length is measured out, and station staves * are fixed at

its extremities. If no such place can be found, the mutual

distance and position of two points conveniently situate for

planting the staves, though divided by a broken surface, are

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