letters, should be of equal lengths; and they should be considerably extended, as this also tends to promote freedom of hand. If the Pupil can acquire the method of writing his long letters with ease and accuracy, he will find no difficulty in making short tops and bottoms, when it is thought necessary to have the lines closer to each other.

Pupils should always have good and correct Copies placed before them; and they should never be allowed to imitate incorrect ones; as this would infallibly lead them into a bad style of writing.-Well and correctly written Copies, we prefer, particularly for Beginners; as they generally feel a greater emulation in imitating them, than they do when writing after Copper Plates.-Of the latter, however, many excellent ones have been published; amongst which we may mention those of Butterworth, Scott, Bradshaw, Findlay, Firth, and Smith.

At the commencement of writing, the Pupil should examine very minutely, every stroke, every turn, and every letter in his copy; and he should take much time, care, and pains, until he can form all the letters correctly. When he can accomplish this, he will soon acquire freedom and expedition, from regular practise; and by always endeavouring to make as many letters as possible, without lifting the Pen from the Paper.

Masters should carefully and regularly examine the copybooks, every two or three lines, as the Pupils are writing; and minutely point out their faults, by noticing the uneven and crooked letters, the square or pointed turns, the too slender down strokes, or the too thick up strokes, the want of uniformity in the size or shape of the letters, the unequal distances of the words, and every other irregularity; and thus, in a little time, the Pupils will not only be able to discover their own faults, but will be taught to judge of any style of writing, almost as well as their Teachers.

Ornamental Penmanship should by no means be neglected; thus Printing, German Text, Old English, and the Italian, and the Engrossing Hands should receive a moderate share of attention, both from the Teacher and the Pupils.These Hands are both useful and graceful, in directing Parcels, ornamenting Books, embellishing PLANS, MAPS, &c., &c.

WE shall close our observations on Writing, with the

Directions given by Mr. Butterworth, in one of his Copy Slips, for acquiring a good and free Current or Running HAND.

Avoid lifting the Pen unnecessarily in Currency.
Be very attentive to your Capitals and long Letters.
Cautiously avoid writing after inaccurate Copies.
Direct your Attention wholly to Accuracy and Freedom.
Endeavour to write as fast as you can write well.
Freedom is soon acquired by striking frequently.
Guard against writing in a slow formal Manner.
Hold the Pen loosely, the Top pointing to the Shoulder.
Imprint on your Mind just Ideas of a Current Hand.
Keenly persevere in writing the Capitals with Taste.
Learn to keep your Words and Lines at proper
Distances.

Make all your small Letters free and the same Size.
Neither sit too upright nor stoop ungracefully.
Often practise writing expeditiously upon a Slate.
Practise writing fast, to save Expence, on waste Paper.
Quick Writing is attained by seldom lifting the Pen.
Remember, to write quickly is of great Consequence.
Slow writing cannot answer in Dispatch of Business.
Take Care in writing quickly, to preserve Correctness.
Write frequently without lifting the Pen, philosophy.

EXTRACT FROM MR. NESBIT'S PRACTICAL ARITHMETIC. "ALL Accounts, Bills of Parcels, Invoices, Receipts, Bills of Exchange, Letters, &c., &c., should be written in a good, free, commercial hand, completely devoid of that stiffness which is generally observable in the hand of a School Boy.--Young persons should endeavour to acquire this hand, before they leave School, by writing frequently, and quickly without lines; always taking care to make as many letters as possible, without lifting the Pen. Whole words may frequently be written in this manner, particularly such as men, him, they, theirs, them, with, come, cheer, lime, time, lint, little, mount, mint, mutton, common, fluent, foment, finite, flutter, fulness, and, land, hand, find, mind, kind, food, florid, almond, confine, commend, composed, compound, &c., &c. By this means the letters will be joined to each other; for with the exception of bad spelling, nothing appears more ridiculous than for persons to write a stiff, broken hand, who have had a good Education.-They should also labour to obtain dispatch in making out Bills, and casting Accounts; as this will be found of great advantage, when they have to perform these operations, in Shops or Counting-houses, on the Spur of the Moment."

NOTE." In writing a running hand the dees, in the middle, and at the end of words, are most easily made, and look most free, when their tops are thrown or struck, with a loose and careless hand."

MATHEMATICS.

MATHEMATICS have been defined by some Writers, "The Science of Quantity;" but it is more properly the Science of Ratio; for it is not the Quantities themselves which are the Subjects of Mathematical Investigations, but the Ratio that different Quantities of the same Kind, have to each other.

In Fact, this Science is a Methodical Concatenation of Principles, Reasonings, and Conclusions, always accompanied by Certainty, as the Truth is always evident; an advantage that particularly characterizes accurate Knowledge, and the true Sciences, with which we must never associate Conjectures, or even Probabilities.

The Subjects of Mathematics are the Comparisons of Magnitudes; as Numbers, Distances, Velocities, Powers, &c., &c.-Thus Arithmetic treats of the relative Values of Numbers; Geometry, considers the relative Magnitudes or Extensions of Bodies; Astronomy, the relative Distances and Velocities of the Planets; and Mechanics, the relative Powers or Forces of different Machines, &c., &c.; in all Cases, some determined Quantity being first fixed upon, as a Standard of Measure.-Quantity expressed by Numbers, is called ARITHMETIC; but when expressed by Numbers, Letters, Signs, and Symbols, it is denominated ALGEBRA.

THE SCIENCE OF MATHEMATICS is divided into a Number of different Branches, as Arithmetic, Geometry, Mensuration, Gauging, Land Surveying, Trigonometry, Navigation, Astronomy, Algebra, Fluxions, Mechanics, &c., &c.-Each of these Branches has its particular Use; and is called into Requisition in conducting the various Departments of the Practical Affairs of Civilized LIFE.

We shall make a few General Observations on each Branch of the Mathematics; and refer our Readers to the Works mentioned at the End of this Essay, in which they will find all the PRINCIPLES and PROPERTIES clearly laid down, and fully explained and illustrated; with ample Directions for their Practical Application to the various Purposes of LIFE and BUSINESS.

These Valuable and Scientific Works have been written by Men of the First Rate Talent and Abilities; and to our Honour as a Nation, chiefly by our own Countrymen; and whoever will make himself Master of these Works, or even of a Part of these Works, may not only become a useful and

an honourable Member of Society, but will rank high, as a MATHEMATICIAN, and as a PHILOSOPHER.

GENERAL OBSERVATIONS.

ARITHMETIC treats of the Nature and Properties of Numbers; and the various Methods of performing Calculations by them.-An Accurate and a Practical Knowledge of this Science, is required in all Trades and Professions; and it is the Basis and Foundation of every other Branch of the Mathematics.

BOOK-KEEPING being a Branch of Arithemetic which is of great and vital Importance, in this Commercial Country; every Exertion should be made to explain and elucidate its Principles and its Practice, in the most clear and perspicuous Manner; particularly as relates to the Nature of Debtor and Creditor, Interest Accounts, Commission, Discount, Profit and Loss, Current Accounts, Exchanges, &c., &c.

Some Merchants do not wish their Sons to study any System of Bookkeeping at School, but rather choose to have them instructed in their own Counting-houses; yet we are of Opinion that it is an excellent Method to Let Young Men go through a regular System of Merchants' Accounts, previously to the completing of their Education.

If a Pupil be properly instructed in School, and made fully to understand Dr. Kelly's Work, or Mr. Morrison's, or any other good modern Treatise on the Subject, he will be able to Form Sets of Books for himself, to suit his own Business; and will very readily comprehend any of the various systems of Book-keeping that may be practised in different Counting-houses; as all these systems have the same Basis; and they all tend to the same Object; namely, that of exhibiting the True State OF A MERCHANT'S AFFAIRS.

The chief objection raised, by Commercial Gentlemen, against Schoolroom Book-keeping, is the imperfect manner in which it is frequently taught; namely, by directing the Pupils first to copy, verbatim, the Waste Book, afterwards the Journal, and lastly the Ledger; instead of carrying all these Books on at the same time, making all the necessary calculations, Journalizing from the Waste Book, Posting from the Journal, and then Balancing each Account properly, in the Ledger, as it is done in Real Business.

It may be of some advantage for Young Teachers to be reminded, that the Journalizing and Posting should always be performed on slates, or on waste paper, previously to making the entries in the Books; as this gives the Master an opportunity of instructing his Pupils in a proper manner, and prevents many mistakes that will otherwise take place in performing these OPERATIONS.

GEOMETRY and MENSURATION are in great Requisition in the Science of Mechanics; and a perfect Knowledge of these Branches must be obtained by all Persons who are designed for Engineers, Architects, Timber Merchants, Builders, Carpenters, Joiners, &c., &c. ; and indeed, if every Person dipped more into Mathematics, they would find an Advantage; for what Trade, Business, or Profession can be carried on without some Knowledge of Geometry and Mensuration ?-Here we may observe that Euclid's Elements of Geometry is the oldest Work that has been brought down to the present time; he flourished about 300 Years before the Christian ERA.

GAUGING is absolutely requisite in the Excise and Customs; and it is also wanted by Maltsters, Brewers, Distillers, Rectifiers, Spirit Merchants, Wine Merchants, Inn Keepers, Victuallers, Vinegar Makers, Cyder Makers, Soap Makers, Starch Makers, Glass Makers, &c., &c.; and as this Science is required in so many Trades, would it not be advisable for every Young Person to obtain a Knowledge of it; as no one knows what Situation he may be called to fill in his Passage through LIFE?

LAND SURVEYING is of extensive Use in ascertaining the Quantity of Land, and of Course the value of Land; for without Measurements, neither the Occupier nor the Owner would be able to come to any Negociation. Therefore, this Science ought to be Studied at School, by all those who are designed for Land Surveyors, or who expect to become Stewards to the Proprietors of Land, and particularly Stewards to Noblemen.-The Science of Land Surveying is also of great Advantage to Farmers, and to all Agriculturalists.

PLANE TRIGONOMETRY is of great importance in almost every Branch of the Mathematics; but particularly in Land Surveying, measuring Inaccessible Heights and Distances, &c., &c.; and it also forms the Introductory Part, or more properly, the very Basis of Navigation.

NAVIGATION in this Country of Ships, Colonies, and Commerce, is not only of paramount Importance, but absolutely necessary for all those who are designed to conduct our Vessels through the Trackless Deep, to far distant Shores; and it ought, therefore, to form a leading feature in the Education of all Persons who are intended for a MARITIME LIFE, or who have Time and Leisure to devote to SCIENTIFIC PURSUITS.

SPHERICAL TRIGONOMETRY is the Science which treats of the Properties and Relations of Spherical Triangles; and of the Methods of determining their Sides and Angles.-This Science leads us on to Practical Astronomy, or the Resolution of Astronomical Problems, which chiefly relate to finding the relative Situations and Positions of the Heavenly Bodies with Respect to each other; and it is required in both Geography and Navigation, as well as in Astronomy.

By this Science, we find the Latitudes and Longitudes of Places on the Earth's Surface, and thus we are enabled to give greater Accuracy to our Maps and Sea Charts; and accommodate both the Geographer and Navigator.-Mr. Emerson says, "Navigation depends upon Trigonometry.-Surveying and Dialling owe their greatest exactness to it.It is of singular Service in Military Affairs.-The very Foundation of Practical Astronomy.-And furnishes us with the only correct Methods of determining the Geographical Distances of Places on the Earth ;and their several Positions with respect to each other."

ASTRONOMY is a Mixed, Mathematical and Philosophical Science, which treats of the Heavenly Bodies; their Motions, Distances, Dimensions, Magnitudes, Orders, Periods, Orbits, Revolutions, Oppositions, Conjunctions, Eclipses, &c., &c.; and investigates the Causes, Principles, and Laws by which their Motions and other Phenomena are produced and regulated. This is a Wonderful Science, and particularly displays the Wisdom, the Power, and the Goodness of the ALMIGHty Designer of the UNIVERSE!!!

ALGEBRA is a General Method of resolving Mathematical Problems, by Means of Equations; and by its Aid, Solutions may be obtained