from a credulous simplicity, most strongly fortify us against the vanity of scepticism, effectually restrain us from a rash presumption, most easily incline us to a due assent, perfectly subject us to the government of right reason. While the mind is abstracted and elevated from 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 contemplations.

SECT. I.-ARITHMETIC.

Arithmetic teaches the method of computing numbers, and explains their nature and peculiarities. The four first fundamental principles, viz. addition, subtraction, multiplication, and division have always, in a certain degree, been practised by different nations.

1. Numbers, as a science, must, in a great measure, have depended on the advancement of commerce, because arithmetical calculations becoming, then, more necessary would receive a greater degree of attention. Thus arithmetic is, with great probability, supposed to have been of Tyrian or Phoenician invention. From Asia it is said to have passed into Egypt. From Egypt, arithmetic was transmitted to the Greeks; thence, with its improvements, it proceeded to the Romans, and from the Romans it has been dispersed over the modern nations of the world. The symbols or characters of numbers, and the scale of numerical calculations have been considerably diversified in different ages. The Hebrews and Greeks, and after them the Romans, had recourse to the letters of their alphabet for the representation of numbers. The Mexicans adopted circles for cyphers, and the ancient Peruvians coloured knotted cords, called quipos. The Indians are, at this time, very expert in computing by means of their fingers; and the modern natives of Peru are said by the different arrangements of their grains of maize, to surpass Europeans, aided by all their rules.

2. The Arabian or Indian notation, which is now universally practised, was originally derived from the Indians,

and was, in the tenth century, brought by the Moois or Saracens from Arabia into Spain. Its improvements principally consist in its brevity and precision; instead of employing twenty-four characters, only nine digits and a cypher are wanted. The symbols also are more simple, more appropriate, and determined; and therefore the powers of them are less liable to inaccuracy or confusion. With the symbols too, the scale of numerical calculations has been vaned. The first improvement was the introduction of reckoning by tens, which, no doubt, took its rise from the obvious node of counting by the fingers, as that was customary in the primary calculation of every nation except the Chinese.

3. The Greeks had two methods of making the advance of numbers; one on the plan which was afterwards adopted by the Romans, and which is still used to distinguish the chapters and sections of books; and in the other, the first nine letters of the alphabet represented the first numbers from 1 to 9, the next nine so many tens, from 10 to 90. The number of hundreds were expressed by other letters, supplying what was wanting either by other marks or characters, or by repeating the letters with different signs in order to describe thousands, tens of thousands, &c.

4. About the year of Christ 200, a new kind of arithmetic, called sexagesimal, was invented by Ptolemy. Every unit was supposed to be divided into 60 parts, and each of these into 60 others, &c. Thus from 1 to 59 were marked in the common way: then 60 was called a sexagesima, or first sexagesimal integer, and had one single dash over it, as I'; 60 times 60 was called 'sexagesima secunda,' and marked I", &c. .These methods of calculation are continued by astrologers in the subdivisions of the degrees of circles. The decuple, or Arabian scale, substitutes decimat instead of sexagesimal progression, and by this single process removes the difficulties and embarrassments of the preceding modes. Thus the signs of numbers, from 1 to 9, are considered as simple characters, denoting the simple numbers subjoined to the character; the cypher, 0, by filling the blanks, denotes the want of a number, or unit, in that place; and the addition of the columns in a ten-fold ratio, always expressing ten times

the former, leads from tens, according to the order in which they stand, in a method at once most luminous and

certain.

5. For decimal parts we are indebted to Regiomontanus, who, about the year 1464, published his book of Triangular Canons.' Dr. Wallis invented the use of circulating decimals, and the arithmetic of infinites; but the last, and, with regard to extensive application, the greatest improvement which the art of computation ever received, was from the invention of logarithms, the honour of which is due to John Napier, baron of Merchiston, in Scotland, who published his discovery about the beginning of the seventeenth century. Mr. Henry Briggs followed Baron Napier on the same subject. Arithmetic may now be considered as having advanced to a degree of perfection which, in former times, could scarcely have been conceived, and to be one of those few sciences which have left little room for further improvement.

SECT. II.-ALGEBRA.

1. Algebra is a general method of computation in which signs and symbols, (usually the letters of the alphabet,) are used to represent numbers, or any other quantities. This science is a kind of short-hand, or ready way of writing a chain of mathematical reasoning on any subject whatever. It is applicable to arithmetic, geometry, astronomy, &c. and the conciseness and perspicuity with which every proposition can be written in algebraic characters, render this method very superior to the tedious circumlocution necessary in words or figures at length.

2. Mathematicians divide this art into two parts, numeral or vulgar, and literal or specious; the former is that of the ancients, where the unknown or required quantities only are represented by letters or characters, and the known ones by number: the latter, which is the new algebra, is much more extensive, the known as well as the unknown quantities being expressed by letters. The process in both methods is essentially the same; the conditions of the question being, in each, denoted by the connecting symbols. Thus by the symbols used

to denote addition, subtraction, multiplication, division, &c. with a few of the letters of the alphabet, the operation is accurately performed, and in much less. time than it could have been done by the common rules of arithmetic.

Select Books on Arithmetic and Algebra.

Greig's, Bonnycastle's, Walkingame's, and Hutton's Arithmetic; all of which are extant in 12mo. Tate's Arithmetic, however, is the

best adapted to real business.

Fenning's Algebra, 12mo. Bonnycastle's Algebra, 12mo.

SECT. III.-GEOMETRY.

1. Geometry treats of lines, surfaces, and solids, and is the doctrine of extension and magnitude in general. Hence a line, an angle, a circle, and, in short, figures of every size or shape, come under the subject of geometry. What has length and breadth only, is termed a superficies, such as the admeasurement of a board, a table, a field, or a country, to determine its contents, in feet, yards, acres, &c. What has length, breadth, or thickness, is termed a solid; and of whatever size or figure it may be, whether a log of wood, a pyramid, or a globe. Geometry is able to ascertain its number of cubic inches, yards, or miles. Geometry, or the art of measuring, like all other useful inventions, appears to have been the offspring of want and necessity; and to have had its origin in those remote ages of antiquity, which are far beyond the reach of credible and authentic history.

2. Egypt, the fruitful parent of almost all the liberal sciences, is imagined to have given birth to geometry or mensuration. After the overflowings of the Nile had deluged the country, and all the artificial boundaries and land-marks were destroyed, there could have been no other method of ascertaining individual property, than by a previous knowledge of its figure and dimensions. From this circumstance it appears highly probable, that geometry was first known and cultivated by the ancient Egyptians; as being the only science which could administer to their wants, and furnish them with the assistance they required. The name itself signifies properly the art of mea

suring the earth, which serves still further to confirm this opinion; especially as it is well known that many of the mathematicians applied their geometrical knowledge entirely to that purpose; and that even the elements of Euclid, as they now stand, are only the theory whence we obtain the rules and precepts of our present more mechanical practice.

3. The use of geometry in most of the different branches of the mathematics is so general and extensive, that it may be justly considered as the parent of all the rest, and the source whence are derived the various properties and principles to which they owe their existence. Artificers of almost all denominations, are indebted to this invention for the establishment of their several occupations, and the perfection and value of their workmanship. Without its assistance all the great and noble works of Art would have been imperfect and useless. testified his conviction of the importance of geometry, by placing over the door of his academy, an inscription to this effect, "Let no one ignorant of geometry enter here."

Select Books on Geometry.

Plato

Donne's Geometrician, 8vo. Bonnycastle's Geometry, 8vo. After these Simpson's Euclid, 8vo. may be studied with advantage.

SECT. IV.-ARCHITECTURE.

§ 1. Sketch of the History of Architecture.

1. Architecture denotes the art of building in general, though it is chiefly applied to the construction of edifices, appropriated to the purposes of civil life, as houses, churches, and bridges. The excellence of architecture consists in such a regular disposition of the materials employed in an edifice, as give it strength, convenience, beauty, and proportion. In the earliest ages of the world we are informed, that a city was built by Enoch, in order to defend himself and his family against the posterity of Abel, whom Cain, the father of Enoch, had murdered. Immediately after the flood, bricks and slime were, used,