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William Shepherd.

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sive privileges. These may be considered as referring to his
foreign and to his domestic relations. With regard to the
former it may be observed : 1 . That as the dignity and power
of the state is supposed to center in the king, he is endued
with the sole power of sending ambassadors to foreign states,
and of receiving witliin his dominions the ambassadors of fo-
reign potentates. 2. That for tlie same reason, it is also his
prerogative to enter into treaties, and to form alliances with
foreign princes and states. 3. That, according to the prac-
tice of former times, if any of his subjects were despoiled or
aggrieved by strangers, he was empowered to grant them let-
ters of marque and reprisal, which authorized them to redress
their private injury. But, in modern days, tliis private appeal
to force is out of use. The wrongs of individuals are made



REGAL PRIVILEGES. 409

'subjects of national resentment, and become a cause of war;
the proclaiming, prosecuting, and terminating of which, con-
stitutes the fourth and the most important branch of the
royal prerogative, as exercised upon foreign affairs.

In domestic concerns, the regal privileges are so ample,
that, did not experience happily prove that they may be
checked and counterbalanced, it might, upon an enumeration
of them, be supposed that they would soon be heightened
into despotism. For, 1. The king has an unquestionable
right to pur his negative upon all legislative measures pro-
posed by the two <5ther branches of the legislature. 2. As
chief of the military power of the kingdom, he appoints all
officers of the army and the navy, assigns them their stations
in peace and war, and directs all the variety of military move-
ments. He superintends all guards and garrisons, and erects
and maintains all forts, fortresses, and places of arms, in these
realms and their dependent colonies. 3. He is the fountain
of justice, and the general conservator of the peace through-
out the kingdom. He has power to erect courts of judica-
ture, and to execute all sentences therein passed, or to sus-
pend, or, with one exception, to remit their execution at his
pleasure. The ajudges of the high courts of judicature are
also appointed by him ; but with a laudable precaution against
a time-serving spirit on the part of those eminent magistrates,
it is provided by statute, that they shall not be dismissed at
the pleasure of the king, but that they shall retain their places
even in the case of a demise of the crown quamdiu se bene
gesserint. 5. The king is the fountain of honour, office, and
privilege. From him are derived all the various degrees of
nobility, knighthood, and other titles. He can also erect
corporations, and convert aliens into denizens, and " grant
place or precedence to any of his subjects, as shall seem good
to his royal wisdom." 6. As arbiter of commerce, the king
is authorized to institute markets and fairs, and to regulate
tolls, weights, and measures, and the public coinage of the
realm.



410 BRITISH CONSTITUTION.

7. And it may be mentioned as the last, though not the
least of the sources of royal power, that the king is the head
of the English and Irish churches, in which capacity he sum-
mons and dissolves all synods and conventions of the clergy ;
and, in fact, though not in express terms, nominates to all
vacant bishoprics, -and to various other ecclesiastical offices.

This detail of the main branches of the royal prerogative
evinces, that the maxim of the law is not a dead letter, which
states, that " in the king is vested the attribute of sovereignty
or pre-eminence." As a further support of his dignity and
authority, it is a sacred principle of ouT constitutional law,
that " the king can do no wrong." Strange as this principle
may appear on its first enunciation, and directly as it may ap-
pear to tend to despotism, it is, in point of fact, by its appli-
cation, one of the main preservatives of public liberty and of
general tranquillity. For, whilst it is asserted that " the king
can do no wrong," it Is provided, that for all his public acts,
his ministers and advisers are responsible to the nation at
large by the medium of the parliament, and of other legally
constituted assemblies. Hence, whilst due* respect is paid to
the person and the character of the reigning sovereign, the
propriety of his measures may be canvassed with strictness,
and even with jealousy and those popular discontents which,
in more arbitrary governments, would give, rise to all the
horrors of a revolution, are generally allayed by the easy and
simple process of a change of administration.

The House of Peers is composed of the Lords sjNritual,
and the Lords temporal. The former consist of two Archbi-
shops, and twenty-four bishops, who are a kind of representa-
tives of the clergy of England and Wales ; and of four Bi-
shops, who are taken by rotation, from the eighteen bishops of
Jreland. With regard to England, the number of temporal
peers is unlimited. At the period of the Union with Ireland,
A. D. 1800, it was settled, that from the Irish temporal peers,
twenty-eight should be elected by their own body to sit in the
Imperial parliament ; tliat they should retain their seats dur-



HOUSE OF COMMONS. 411

ing life ; and that each vacancy by death, should be filled by a
new election. The Scotch peers are sixteen in number, and
are elected by their own body, for one parliament only. The
Lords temporal are divided into dukes, marquisses, earls,
viscounts, and barons, who hold their respective ranks in the
foregoing order, by hereditary descent or by creation. In its
aggregate capacity, the House of Peers has a right to a nega-
tive upon all legislative proposals. Its members also enjoy
the privilege of voting by proxy, and of entering their dissent
against the measures adopted by the majority, with the reasons
of such dissent, upon the journals of the house. It is the
final court of appeal in civil causes ; and it constitutes a crimi-
nal court for the trial of its own members, and in certain
cases for the trial of others who are impeached of high crimes
and misdemeanours.

The Commons of the United Kingdoms of Great Britain
and Ireland consist, to adopt the language of Blackstone, *' of
all such men of any property as have not seats in the House
of Lords, every one of which has a vote in parliament, either
personally, or by his representatives." These representatives,
which constitute the House of Commons, or as it is some-
times called, the Lower House of Parliament, are returned by
the respective kingdoms in the following proportions :

England - - 513 -
Scotland - - 45
Ireland - - 100



Total - 658 ' '^



The members of the House of Commons are divided into
two classes, Knights of the Shire, or representatives of coun-
ties ; and citizens and burgesses, or representatives of cities
and boroughs. The qualification for voting for the former, is
the possession of a freehold, of the value of forty shillings per
annum, or upwards, situated in the county, for which the



412 BRITISH CONSTITUTION.

elector claims a vote. The right of election in boroughs is
very various, depending upon the charters and immemoiial
usages of each place, or upon solemn decisions made by com-
mittees appointed by the House, to decide upon the merits of
election petitions. With a view to guard the purity, and to
ensure the respectability of this assembly, it is provided by
statute, that several descriptions of place-men and pensioners
of the crown .shall be ineligible to sit in it ; and that each
knight of the shire, must be seized of a freehold estate in the
county which he represents, of the value of o600 per annum ;
and that every burgess and citizen must possess an annual
landed income of 300.

The House of Commons, thus constituted, like the other
branches of the legislature, enjoys the privilege of a negative
upon all the laws which may be proposed for its consideration.
By immemorial custom also, it claims and exercises the right
of originating all money bills ; that is to say, all bills which
levy money upon the subject by way of taxes or assessments.
Some fanciful reasons have been assigned for this privilege;
but, referring to the history of the Constitution of the Lower
House, it should seem, that it originated in the circumstance,
that the Commons were first summoned merely for the purpose
of informing the king what weight* of taxation, their respective
constituent bodies would be able to bear. This provision,
which was at first in all probability regarded as a great hard-
ship, has by the lapse of time been modified and matured, so
as to become the principal safeguard of our liberties, and the
main barrier against any inordinate encrease of the power of
the crown. Without money the strength of the executive is
paralysed ; and as all taxation must not only be approved bj
the Commons, but must originate in their House, while on
the one hand the King and the Lords are armed with a ne-
gative to defend themselves against any attempt at encroach-
ment on the part of the third estate ; so on the other, the
Commons can at any time check the measures of ministerial
folly, or guilt; by withholding the supplies. At the same time,



SUMMARY OF 413

should it ever happen, that the proceedings of the Commons
are manifestly dictated by factious views, which are incom-
patible with the general interests of the community, the kuig
is invested with a power, not only to negative their proceedings,
but to dissolve the parliament ; and thus, by submitting their
conduct to the revision of their constituents, to appeal against
them to the nation at large.

Thus, however complicated the machinery of the British
Constitution and Government may appear to be, its operations
are regulated by a few grand and simple principles ; namely,
the sacredness of the king's person, and the responsibility of his
ministers the negative power of each branch of the legislature;
and the command. exercised by the Commons over the public
purse. The forms of parliamentary proceedings provide for
the most mature discussion, and the greatest possible freedom
of debate ; and by those forms, ample opportunity is given to
the people at large, and even to individuals, to state to either
house, by the means of petitions, their sentiments upon
measures which affect, or seem to affect, their immediate and
peculiar interests. Notwithstanding then the influence of the
crown is occasionally and plainly perceptible in the proceedings
of the other branches of the legislature, the constitution has pro-
vided a check to that influence by the counterbalance of pub-
lic opinion. The maxim of the great lord treasurer Burleigh
is strictly true, that " England can never be ruined but by a
parliament ;" but it is equally true, that no parliament will
hazard any proceedings ruinous to tlie liberties, or to the
general welfare of the country, while the people at large
watch with due vigilance the conduct of their representatives.



>Mf'



CHAP. XXVIII.



MATHEMATICS.



Objects of Mathematics Use, importance and history of Matheniaticit.
Arithmetic; history of importance of to other branches of science
books of Arithmetic, viz. those by Dilworth Walkinghame Vyse Hot-
ton Molyneux Bonnycastle Joyce Malcolm and Mair Arithmeti-
cal machines.

1 HE science of mathematics treats of the ratio* and com-
parison 'of quantities, and has been sometimes denominated
the science of ratios. The term mathematics is derived from
a Greek word signifying discipline, or science, thereby sig-
nifying the high idea that we ought to entertain of this branch
of human knowledge. The science of mathematics is always
accompanied v/ith certainty, an advantage that character-
izes accurate knowledge, and the true sciences, with which
we must on no accoimt associate conjecture, or even pro-
bability.

* Ratio is the relation that two magnitades of the same kind bear
to one another in respect to qnantity. Thus the ratio of 2 to 1 is double,
of 3 to 1, triple. Ratio has been frequently confounded with proportion,
but they are different things, for proportion is die similitude or equality of
two ratios. Thus the ratio of 6 to 2 is the same as that of 3 to l, and the
ratio of 15 to 5 is tlie same as that 3 to 1, and therefore the ratio of 6 to 2
is the same with that of 15 to 5, which constitutes proportion, and it is thus
expressed, 6 is to 2 as 15 to .S, or thus 6 : 2 : : 15 : 5. So that ratio
exists between two terms, but proportion between two ratios or four temu.



ADVANTAGE OF MATHEMATICS. 415

The subjects of which the mathematics treat, are the compa-
risons of quantities and magnitude, as numbers, velocity, distance,
&c. Quantity expressed by numbers is called Arithmetic ; but
when expressed by letters and signs, it is denominated Algebra.
Geometry considers the relative magnitude and extension of
bodies : astronomy, the relative velocities and distances of the
planets : mechanics, the relative powers and force of different
machines, &c. In all cases there must be some determinate
quantity fixed on as a standard measure.

Mathematics are usually divided into the pure or abstract,
and the compound or mixed. Pure mathematics relate to
magnitudes generally and abstractedly, and are founded on
the elementary ideas of quantity : such is Arithmetic, or the
art of computation, and such also is Geometry, or the science
of mensuration and comparison of extensions of every kind.
Mixed .mathematics consider quantity as subsisting in mate-
rial substances, as length in a pole or staff, depth in a river,
height in a tree, tower, 8cc. Mixed mathematics are very ex-
tensive, and are distinguished by various names, according to
the different subjects they consider^ and the different views in
which they are taken, such as mechanics, hydrostatics, hydrau-
lics, acoustics, optics, navigation, &c.

There is an advantage m pure mathematics, which is pecu-
liar to that branch of science : it neither occasions nor admits
of contests among wrangling disputants, as happens in othe
branches of knowledge. For the definitions of the terms are
premised, and every person that reads a proposition has the
same idea of every part of it. Heiie, it is easy to put an
end to all mathematical controversies, by shewing, either that
the adversary has not kept to his definitions, or has not laid
down true premises, gr has drawn false conclusions from true
principles ; and in case some one or other of these can be
done, he must acknowledge the truth of what has been proved.
In mixed mathematics, where we reason mathematically upon*
physical subjects, our definitions cannot be so strict as in geo-
metry. They are.frequently little better than descriptions, which.



416 MATHEMATICS.

however, with proper care, will be nearly of the same use as
definitions, provided we always mean the same thing by those
terms which we have once explained.

Though tlie application of mathemati( al sciences is of great
importance to the interests of society, yet this is not that
which constitutes their pecuhar excellency ; it is their opera-
tion upon the mind, the vigour which they impart to our
intellectual faculties, and the discipline which they impose
upon our rational powers. The eulogium passed upon this
science by the learned Dr. Barrow, may claim the notice of
every youth ; and those who are capable of appreciating the
value of this branch of knowledge, will admit, that he has not
gone, in any respect, beyond the truth. " The mathematics,"
says the doctor, " effectually exercise studious minds, and
plainly demonstrate every thing witliin their reach : they draw
certain conclusions, instruct by profitable rules, and unfold
pleasant questions. These disciplines inure and corroborate
the mind to a constant diligence in study : they wholly deliver
us from a credulous simplicity ; and most strongly fortify us
against the- vanity of scepticism : they effectually restrain us
from rash presumption, most easily incline us to a due assent,
and perfectly subject us to the government of right reason.
While the mind is abstracted from and elevated above sensible
matter, it distinctly views pure forms, conceives the beauty
of ideas, and investigates the harmony of proportions; the
manners themselves are sensibly corrected and improved, the
affections composed and rectified, the fancy calmed and settled,
and the understanding raised and excited to more divine con-
templations."

By a certain class of people, mathematical studies are held
in contempt; but it may, perhaps, be justly assumed, that those
who speak, or even think slightly of them, are the persons who
have not patience to investigate the truths which they are
capable of unfolding.

A liberal mind excludes, as unworthy of its notice, none
of the sciences. They all contribute, by various, and, perhaps,



ADVANTAGE OF MATHEMATICS. 417

different means to adorn and embellish life, and for this rea-
son, independently of the advantages above enumerated, they
ODght to be cultivated and improved. " Happy is the mind
that is not contracted by the study of philosophy, nor ener-
vated by the charms of the Belles Lettres ; that can be
strengthened by Locke ; instructed by Clarke and Newton ;
impassioned by Cicero and Demosthenes ; and elevated by the
powers of Homer and Virgil."

The limits of our work, will not allow us to enter at large
into the history of any branch of science; we may, however,
briefly notice some facts connected with the early culture of
mathematical knowledge. It is generally supposed that the
Greeks derived their scientific learning from the Magi of
Egypt, long before the time of Thales, who is often styled the
father of the Grecian philosophy, only because he is the first
of those of whom any decided account has been transmitted to
us. This philosopher, according to Herodotus, predicted a
total eclipse of the sun, which from subsequent inquiry is as-
certained to have happened in the year 610, B.C. At this
period, then, astronomy must have made considerable progress
in Greece, as the prediction of an eclipse requires much pre-
vious knowledge, and a vast number of accurate observations,
which could only have been obtained in a long series of years.
Pythagoras, who was probably a pupil of Thales, and who
flourished about the year 590, B. C, had made considerable
improvements in arithmetic, astronomy, and geometry. In
arithmetic, he invented the multiplication table, or the Abacus
Pythagoricus : in astronomy, he suggested the idea of the
true system, placing the sun in the centre, and making the pla-
nets, revolve about that body ; ^nd in geometry, he discovered
the forty-seventh proposition of Euclid's first book, well known
by the name of the Pythagorean theorem, and which of itself,
would have conferred a sort of immortality upon the disco-
verer. At or about this period, Anaximander flourished, and
soon after Anaximenes, Anaxagoras, and others of great note :
but passing over these, we come to Plato, who cultivated both

VOL. I. 2 E



418 MATHEMATICS.

geometry afid astrunoniy with great assiduity^ nearly four ceii-
turieii before tlic birth of ChrUt. To him, and to his disciple
Aristasm, we are indebted for the introduction of the conic
sections in geometry. It was almost a century between the
time of Plato and Euclid, daring which period all the sciences
were considerably advanced and extended ^ and treatises on
particular subjects appeared from this time, in which all the
propositions then known, were collected and arranged iji sy*-
temalic order, whicli was the object of Euclid in his cele-
brated Elements, a work of which we shall shortly have occa-
sion to speak, and which has met with a success vastly supe-
rior to that of any other book of science that ever was pub-
hshed. Archiiuedes, one of the most illustrious geometricians
tha^evei- existed, followed soon after the time of Euclid. His ge-
nius was as extensive as it was penetrating, and it led him to the
contemplation of almost every species of human knowledge,
and nearly every branch of mathematical science is indebted to
him for most important discoveries. After Archimedes, at
the distance of tifty years, ApoUonius cultivated the matliema-
tical sciences with the greatest possible success. About this
period flourished Eratostlieues, who hrst attempted to meaMire
the earth ; Ctesibius, to whom we are indebted for pumps ;
and Hero of Alexandria, who invented the clepsydra.*, or
water-clocks. Hipparchus, the prince and father of astronomy,
who flourished about one hundred and forty years, B.C., made
the fust classification of the stars, ascertained nearly tlie du-
ration of the year, discovered the eccentricity of tlie earth's
orbit, the precessiou of the equinoxes, ami applied his favou-
rite science to the purposes of geography.^* Theodosius is the
next eminent mathematician ; he wrote a ^ork on tlie sphere,
which may be reckoned as the foundation of spherical trigono-
metry ; but after him, we have no account of any great geo-
metrician for nearly three hundred years. Ptolemy began to
revive tlie sciences, about a century and half after the birth of
Christ. His principal work was entitled the Almagest, au
Arabic word signifying " the Great Collection/' because it con/-



HISTORY OF MATHEMATICS. 419

tained all the ancient observations and theories, \vhich, with
the addition of his own researches, may be said to have formed
the most complete collection of ancient astronomy that ever
appeared. This work, in some measure supplies the place of
those that preceded it, and which are now lost to the world ;
and fxjr the compilation of it, Ptolemy is entitled to the grati-
tude of all lovers of science. Some years after this period,
Diophantus made a new and remarkable step in arithmetic, by
the invention of the indeterminate analysis, a species of alge-
bra, which is the first trace we have of this extensive and
highly useful branch of science. This work it appears, con-
sisted of thirteen books, only six of which have come down to
us : it displays great talents, and has ever been in high esteem
among analysts of all succeeding ages ; and has accordingly
been commented upon and explained by various writers, both
ancient and modern most of the former are lost. Of these,
mathematicians have chiefly regretted the commentary of
Hypatia, daughter of Theon, who flourished about the four
hundred and tenth year of the Christian ara. The talents,
virtues, and misfortunes of this illustrious victim of fanaticism,
have a claim to the homage of posterity, while the remem-
brance of the deed,* and the perpetrators of it, will be exe-
crated and abhorred by every friend to science, and admirer of
female virtue and talents. About this period we meet with
Pappus, who is distinguished for his collections of the various
works of his predecessors : these collections contain one of
the most valuable monuments of ancient geometry. In them,
he has assembled together a number of excellent works, the
originals of almost all of which are now lost, and to them he
has added several new, curious, and learned propositions of
his own.

* Hypatia was murdered in tlie year 413, under. the reign of Theodosius
II. by the consent and desire, if not by the direct instigation of Cyril, a
Christian bishop. Her cruel and blood-thirsty enemies were not content
with her life : they put her to extreme torture, and then treated her dead
body with the utmost indignity.

2e 2



420 MATHEMATICS.

After Pappus, Eutochiiis flourished about the year 520,
who was a great mathematician, and whose commentaries on
the works of Archimedes and Apollonius are in high estima-
tion. Other names might be enumerated, had we room to
dilate ; but we pass on to that period which proved so fatal
to the sciences. These had for a long time taken refuge in
Alexandria ; but, about the middle of the seventh century, a
tremendous storm arose, which threatened their total destruc-
tion. The successors of Mohammed ravaged an immense ex-
tent of the then civilized world. All the cultivators of the
arts and sciences, who, from every nation, had assembled in
Alexandria, were driven away with ignominy ; some fell be-
neath the swords of the conquerors, others fled into remote
countries, to drag out the remainder of their lives in obscurity
and distress. The places and instruments which had been so
useful in making an immense number of astronomical observa-
tions, were involved with the records in one common ruin.
The whole of that precious library, which contained the works



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 34 of 44)