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William Shepherd.

Systematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) online

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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



Online LibraryWilliam ShepherdSystematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) → online text (page 43 of 44)