afford you, I do not despair of success, if only you will strictly observe my requirements. I shall, however, have little chance of carrying my wishes into effect, if you begin with a notion that the task is an easy one. Were it an easy task, it would not be worth your trouble. I tell you plainly that the task is hard; that it is a long task; that it will require on your part patience and diligence. If you commence the study under the delusion that it is mere play, you will do as very many have done before you,-in a short time you will grow weary, and give it up. But if, in a true manly spirit, you take up the task as a worthy task, as a useful though a difficult task, as a task in which whatever I may do for you, you must do far more for yourself, then you will not fail to make progress; your course will become easier as you go on; and you will have the delightful satisfaction that you are employing your time and your faculties so as to produce definite, important, and abiding results. If, however, it is not an easy thing to acquire an acquaintance with the principles of the Latin language, then you will at once see that it is not wise to be engaged at the same time in other difficult studies. Many persons, especially young persons, fail to give themselves a good education, because they attempt too much-because, indeed, they attempt what is impossible. I remember that when I began in earnest to study Latin, I was saved from this danger, by the judicious opinion of a learned and venerated friend, who advised me not to attempt more than one subject at a time. In part, my friend, I am like you, self-taught. In the ordinary phrase, indeed, I learnt Latin when a boy at school; but so wretched were the methods of teaching some forty years ago, that though I learnt Latin I knew nothing of Latin when I left school. Soon, however, 1 began the study in earnest; I began almost unaided. But an adviser came, and under his directions I for a time confined myself almost exclusively to my Latin studies. They were hard, very hard; much harder were they to me than they need be to you. But by giving my mind perseveringly to the task, I reaped success, and so in time became prepared for college. Let me be your adviser. Even in the treasures of the POPULAR EDUCATOR you may find a source of danger. If you engage in all the studies here offered, you will become proficient in none. Make a selection. If you wish to learn Latin, pursue the study wisely. But wisely you cannot pursue it, if you combine therewith several other subjects. For you, one language at a time is enough, Do not attempt more. If you want relief from this, which is a severe study, unite with it the Lessons in History which these pages supply, But do not attempt to learn French while you are learning Latin, When you know Latin, you will find French very easy. With a knowledge of Latin you would also have no difficulty at all in becoming, in a short time, acquainted with the Italian, the Spanish, and the Portuguese languages, which have Latin for their common parent. In the Latin, too, you would find a great assistance should you wish to learn Greek, and so acquire the power of reading the scriptures of the New Testament in their original tongue. You may practically regard the Latin alphabet as the same as the English. The English letters may be traced to the Latin; the Latin letters are derived from the Greek; and the Greek are in substance identical with the Phenician; while the Phenician alphabet is the oldest, or one of the oldest, in the world, In the pronunciation, too, you may in the main follow the best English usage. Every modern nation pronounces the Latin as it pronounces its own tongue. Thus there are divers methods of pronunciation. This diversity would be inconvenient, if the Latin were, like the French, a general medium of verbal intercourse. At one time it was so. And then there prevailed one recognised manner of pronunciation. Now, however, for the most part, Latin is read, not spoken. Consequently the pronunciation is not a matter of consequence. Even in our own country there are diversities. but such diversities are secondary matters. To one or two remarks, however, you should carefully attend. In Latin the vowels are what is called long or short. In other words, on some the accent or stress of the voice is thrown, on others it is not thrown. The vowel a for instance is mostly long, the vowel is mostly short. A long vowel is said to be equal to wo short vowels. We English people, however, have no other way of marking a long vowel, except by throwing on it, the accent or stress of the voice. It is also a fact, that the same vowel is sometimes short and sometimes long; in other words, the same vowel sometimes has, and sometimes has not the accent on it; thus the i in dominus, a lord, is without the accent, while the i in doctrina, learning, has the accent; the former, therefore is pronounced thus, dóminus, the latter thus, doctrína. Now observe that these words are trisyllables, as dom, i, nus. Of these three syllables the last, namely us, is called the ultimate; the second, i, is called the penult; the third, or dom, is called the antepenult. And the general rule for pronouncing Latin words is, that the accent is thrown on the penult, or if not on the penult, then on the antepenult. In doctrina the accent is on the penult, or last syllable but one. In dóminus, the accent is on the antepenult, or last syllable but two. In order that you may know where to lay the stress of your voice. I shall mark, as in dóminus and doctrína, on which syllable the accent lics. You will then understand that when I put a mark thus over a vowel, I mean thereby that you should let your voice rest, as it were on that vowel. For example, in the word incur, the accent you know is on the last syllable, for you throw the stress of the voice on the syllable cur. This is indicated thus, incúr. So in the Latin amicus, a friend, the accent is on the i, and the word is to be pronounced thus amicus, the accent being on the penult. There is another way of marking the same fact; it is by the use of a short strait line, as, and a curve, as . The former denotes a long or accented syllable, for instance doctrina; the latter denotes a short or unaccented syllable, for instance, dominus. You thus see that doctrina and doctrína, dóminus and dominus point out the same thing, namely, that in pronouncing doctrína you must lay the stress of the voice on the i, and in pronouncing dóminus you must lay it on the o. I must point out to you another practice. In Latin, as you will presently learn, the endings of words have a good deal to do with their meanings. It is, on that account, usual to pronounce them at least very distinctly. Indeed, I might say, that on every terminating syllable a sort of secondary accent is laid. Thus, dominus is pronounced dóminus. So in other forms of the word, thus, dóminí, dóminó, dóminúm. The object is to mark the distinction between, say, dominus and domino, a distinction of great consequence. Another form of this word is dominos. For the same reason a stress is laid on the termination os, which accordingly is pronounced as if it were oase. Words, too, which end in es have a secondary accent on the e; as Vulpes, a jor, pronounced vulpecs. In a few cases the vowel is what we call doubtful, that is, it is sometimes short and sometimes long. This peculiarity is marked thus, as in tenebrae, darkness, when the accent may be on the penult, as tenebrae, or on the antepenult, as tenebrae. Observe, also, that a vowel at the end of a word is always pronounced in Latin. Take, as an example, docéré, to teach, which is pronounced as it is marked, that is, with an accent on the last The Latin syllable no less than on the last syllable but one. language has no silent e, as we have: for instance, in wife. Practise yourself, according to these rules, in pronouncing thus the opening lines of that fine poem, Virgil's Aeneid. As I am its meaning, I subjoin the translation made by the English anxious that you should not pass anything without knowing poet Dryden. "Arma virúmque canó, Trójaé quí prímus ab óris In pronouncing the third line, you must cut off the um EXERCISE-ENGLISH-LATIN. before the voweli; and the e in ille before the e in et. Also Find English words derived from some part of curro; find Engin the fifth line drop the e in quoque before thee in el. In lish words derived from curro, with in prefixed; also with con the last line, too, the e in atque is dropped or elided before prefixed; also with dis prefixed; also with ex prefixed. the vowel a in altae; pronounced as if written qualtae. Accuracy of pronunciation, however, is not easily acquired from any written or printed directions. The living tongue is the only adequate teacher. And it will be well if you can get some grammar school-boy to read to you and hear you read the passage I have given above from Virgil, and the exercises, or some of them, which ensue. Although the pronunciation of Latin is of secondary importance, yet you must try to be as correct as you can, if only from the consideration that what is worth doing at all, is worth doing well. But should you, as you justifiably may, hope by these lessons to prepare yourself for becoming even a teacher of Latin-say in a school-you would in that capacity find the pronunciation considered as a matter of consequence; indeed a disproportionate value is, especially in the old grammar schools, attached to the established methods of pronunciation. After all, we cannot pronounce the Latin as it was pronounced by the Latins them-nature of language itself, as well as receive good mental dis selves, nor can the best trained lips pronounce their poetry so as to reproduce its music. In regard to the exercises which I am about to give, you should first learn the vocabulary by heart. If yours is a mechanical trade, you may repeat the words over again and again while engaged in labour. Or you may make the words your own while walking to and from your employment. Among my personal friends, is a gentleman who acquired the greater part of the words of the French language, while rising and dressing in the morning. Thousands of words have myself learnt while walking for recreation. I Having thoroughly mastered the vocabulary, take a slate and write down the Latin into English; then write the English into Latin. Look over what you have done carefully. Correct every mistake and error. If you look into the exercises you will find that the English will assist you in writing the Latin, and the Latin will assist you in writing the English. When you have got both the Latin and the English into as correct a state as you can, copy them neatly into a note-book. Having done so, read them carefully over, and compare each instance with the rule or the direction, and also the example. Leave nothing until you understand the reason. All the examples or illustrations that I give, as well as the chief rules, should be committed to memory. Before you proceed to a second lesson, ascertain that you are master of the first. It would be useful to write out the rules in one consecutive view, in order that, having them all at once under your eye, you may study them in their connexion and as a whole, so as to see their bearing one upon another, and the general results to which they lead. Such a practice would have a very beneficial effect on your mind, by habituating it to arrangement and order, and might be expected to afford you valuable aid, both in other studies and in your business pursuits. Carefully avoid haste and slovenliness. Do your best in all that you undertake. "Well" not "much" should be your watchword, Repeated reviews of the ground passed over are very desirable. Every Saturday you should go carefully over what you have done during the week. At the end of every month the work of the month should be reviewed. On arriving at a natural division of our subject, as for instance, when we have treated of the nouns, you should go over, and put together in your mind the substance of what has been said thereon. "Be not weary in well doing, for in due season you will reap, if you faint not. (Gal. vi. 9). REMARK.-In order to make my meaning quite clear, I will From cursus comes the myself do this exercise in part. English word course; from in and curs comes incursion; from ex and curs comes excursion. If the reader is acquainted with, or is learning French, he will do well as he passes on, to find out French words corresponding to, and derived from, Latin By words; as in courir, French to run; cours, a course. comparison he may occasionally find that the same sound or word has a different meaning in French from what it has in Latin or in English. Thus, concursus in Latin means a coming together, as to a meeting, a concourse of people; but the corresponding French, concours, signifies co-operation. So concurrence in English is agreement, but in French competition. By practising comparisons such as this, you will not only meet with many curious facts, but be assisted to understand the cipline. If it seems strange to you that the same letters curr or curs should bear dissimilar meanings, a little reflection curr. Its primary meaning is to run. Now, men may run will take away your surprise. Go to the primary meaning of into, or run out of, or run together, or run about, for different purposes. For instance, they may run together in harmony, and then they concur; or they may run together in rivalry, and then they are in what the French call concurrence, that is, competition. immense field. It is only a hint or two that I can give; but I have thus, my fellow student, opened out before you an if you follow these intimations, you will in time become not only a Latin scholar, but a good linguist. LESSONS IN GEOMETRY.-No. I. THE term Geometry, which comes from the Greek word Geome- The word geometry, used in its highest and most extensive meaning, signifies the science of space; or that science which investigates and treats of the properties of, and relations existing among, definite portions of space, under the abstract division of lines, angles, surfaces, and volumes, without any regard to the physical properties of the bodies to which they belong. In this sense, it appears to be very doubtful whether the Egyptians or Chaldeans knew anything of the science. It is to the Greeks, therefore, that we must look for the real origin of geometry, as an abstract science. Thales, the Greek philosopher, born 640 B.C., is reported, by ancient historians to have astonished even the Egyptians by his knowledge of this science, The founder of scientific geometry in Greece, however, appears to have been Pythagoras, who was born about 568 B.C. He discovered the celebrated 47th proposition of the first book of Euclid's Elements, and various other valuable and important theorems. He was great also in astronomy, having anticipated the Copernican system of the world. Of Plato, another great geometrician, and founder of the academy at Athens, we have already spoken in our first number. He was the first who made some advances into what is called the higher geometry. The next name, super-eminent in the science of geometry, is that of Euclid, whose Elements have been the principal text-book for learners, during a period of more than 2000 years. He flourished at Alexandria, in Egypt, about B.c. 300, during the reign of Ptolemy Lagus, who was one of his pupils, and to whom he made the celebrated reply, when asked if there was a shorter way to geometry than by studying his Elements :-"No, sire, there is no royal road to geometry." subject, and others connected with it, we cannot now enter; but we trust to be able, in future numbers of this work, to bring before our readers both its history and its application, as one of the greatest of our modern engines in the discovery of scientific truth, and in the development of the philosophy of nature. DEFINITIONS. Extension, or the space which anybody in nature occupies, has three dimensions, viz., length, breadth, and thickness. This is Euclid's definition of a geometrical solid. A point is the beginning of extension, but no part of it; hence it is said to have position in space, but no magnitude. A line is extension in one direction only; hence, it is said to have length without breadth. Hence, also, the extremities of a line are points; and lines intersect or cross each other only in points. In giving our first lessons on geometry, we think it advisable to follow what seems to have been the natural course of events in the history of this science. The present advanced state of our geometrical knowledge was preceded in early times by a species of practical geometry gathered from experience, and suited to the wants of those who required its application, before any attempt was made to enter very deeply into the study of the theory. The latter was left to the schools of the philosophers and the academy of Plato. Accordingly, we shall precede our disquisitions on the Elements of Euclid and other geometers, both ancient and modern, by a short system of practical rules and easy explanations in this important science; and we shall endeavour to make the subject both The prince of ancient mathematicians, however, was the simple and clear by plain definitions, suitable diagrams, and celebrated Archimedes, born at Syracuse, B.c. 287, about the palpable demonstrations, after the manner of the French period of the death of Euclid. His discoveries in geometry, writers on this subject, who have even in their more elaborate mechanics and hydrostatics, form a remarkable era in the his-treatises to a great extent abandoned the system of Euclid. tory of the mathematical sciences; and even the remains of his works which are still extant, constitute the most valuable part of the ancient Greek geometry. He was the first who attempted to solve the celebrated problem of the rectification of the circle; that is, finding a straight line exactly equal to the circumference. He found out the beautiful ratios of the cylinder to its inscribed sphere and cone, and the quadrature of one of the conic sections. We have, in our second lesson on arithmetic, alluded to his discoveries in that science. His discoveries in physics or natural philosophy are simple, true, and beautiful. The story of the determination of the specific gravity of the golden crown of his cousin, Hiero king of Syracuse, is well known; and the very natural shout of "Heureka, heureka!"—I have found it, I have found it! on coming out of the bath, has become a "household word." Scarcely less celebrated was the famous Apollonius of Perga, in Pamphylia, who flourished at Alexandria in the reign of Ptolemy Euergetes, (from B.C. 247 to 222) another king of the same Ptolomean dynasty, and who was called by his cotemporaries the "Great Geometer." He wrote several books, full of discoveries, on the higher geometry, and greatly extended the domains of the plane geometry. Other geometricians of eminence arose in the school of Alexandria, and bequeathed the precious remains of their genius to happier times. Claudius Ptolemæus, the author of the great work on astronomy called Megale Syntaxis, the Great Construction, or Almagest; Pappus, the author of the Mathematical Collections; and others, including Theon and his daughter Hypatia, bring us down to the period when the Alexandrian library was burnt by command of the Mohammedan barbarian Caliph Omar, and the labour and learning of ages were irrevocably destroyed. The dark ages supervened, and little was done in the advancement of science until the glorious invention of printing, and the general revival of literature about the middle of the fifteenth century. The ancient Greek geometry was speedily made known to the moderns through the medium of translations of, and commentaries upon, the writings of the great masters. The Elements of Euclid, indeed, were reckoned so perfect, that no attempt was made to supersede them; and the only object of writers on geometry in general was to explain his works, and to make what additions they could to the science, in the same masterly style of composition. A host of names of eminent authors might be mentioned, who succeeded in establishing the Greek geometry, and in extending its domains. The principal of these, however, was Dr. Robert Simson, professor of mathematics in the University of Glasgow, who flourished in the middle of the last century. His grand endeavour was to present to modern Europe the Elements of Euclid, as they originally appeared in ancient Greece. In this he succeeded to admiration, and his edition of this great work maintains its reputation to the present moment. He was also an original writer of great eminence; and but for the eclat of the new geometry, invented by Leibnitz and Newtor, he would have shone as a star of the first magnitude. On this interesting  straight line is said, by Euclid, to be that which lies evenly between its extreme points; and, by Archimedes, to be the shortest distance between any two points. Both of these definitions are defective; the defect is supplied thus: A straight line is such, that if any two points be taken in it, the part which they intercept (or which lies between them), is the shortest line that can be drawn between those points." A crooked line is one composed of straight lines joined at their extremities in any manner whatever, except that of uniform direction. A curved line, or curve, is a line whose direction varies at every point. Straight lines, or curve lines, are generally denoted, in speaking and writing, by two letters placed commonly at their extremities; but, they may be placed anywhere on the lines Fig. 1. at a distance from each other. Thus, in fig. 1, the letters A, B, denote one straight line; the letters C, D, another; and the letters E, F, a third; and these straight lines are respectively called the straight lines A B, c D, and E F. A straight line, as A B, may be divided into any number of equal parts, to serve as a standard for measuring other straight lines. A combination of straight, crooked, and curved lines is represented in fig. 2; A B, BC, C D, and D A, are each straight lines; the combination ADC B, beginning at A, and terminating at B, is a crooked line; and the line A M B, beginning at A, and ending at B, is a curved line. A Fig. 2. M B C D being taken, the straight line between them lies wholly in that Parallel straight lines are such as lie in the same plane. and which though produced ever so far both ways, Fig. 3. do not meet (fig. 3). straight lines o A, o B, OC, O E, is a radius, and the straight line A B is a diameter. An arc of a circle is any part of its circumference; the chord of an arc is the straight line which joins its extremities. A segment of a circle is the surface inclosed by an arc and its chord. A sector of a circle is the surface inclosed by an arc, and the two radii drawn from its extremities. Thus, in fig. 7 the portion of the circumference AM C, whose extremities are the same extremities is also an arc; the straight line A c is th chord of either of these arcs. The surface included between the arc A M C and its chord a c, is the segment A MC; there is also the segment A B C. The surface included between the radii o c, o B, and the arc c B, is called the sector coв; the remaining portion of the circle is also a sector. An angle is the inclination of two straight lines to each other, which meet in a point, and are not in the same direction. The point in which they meet is called their vertex, and each of them is called a side or leg of the angle. The angle itself is generally called a plane rectilineal angle, because it necessarily lies in a plane, and is formed of straight lines. Cur-A and c, is an arc; and the remaining portion A B C, having vilineal angles are such as are formed on the surface of a sphere or globe; but the consideration of such angles belongs to the higher geometry. The magnitudes of angles do not depend on the lengths of their legs or sides, but on the degree or amount of aperture between them, taken at the same distance from the vertex. An angle is generally represented by three letters, one of which is always placed at the vertex, to distinguish it particularly from every other angle in a given figure, and the other two Fig. 4. are placed somewhere on the legs of the angle, " but generally at their extremities; and in reading or in speaking of the angle, the letter at the vertex is always placed between the other two, and uttered or written accordingly. Thus, in fig. 4, which represents an angle, the name of the angle is either BA C or CAB: the point A is called its vertex; and the straight lines B A, C A, its Bides or legs. B Angles are divided into two kinds, right and oblique, and oblique angles are divided into two species, acute and obtuse. When one straight line meets another, at any point between its extremities, and makes the adjacent or contiguous angles equal to each other, each of them is called a right angle, and the legs of each of these angles are said to be Fig. 5. perpendicular to one another. Thus, in fig. 5, the straight line A B meets the straight line o D in the point A, and makes the adjacent angles CA B, DA B, equal to each other; each of these angles is therefore called a right angle; and the straight line A B is said to be perpendicular to the straight line A C, or DA, and consequently a C or A D is perpendicular to A B. B с A semicircle is the segment whose chord is a diameter. Thus in fig. 7. A C B, or A E B is a semicircle. The term semicircle, which literally means half-a-circle, is restricted in geometry to the segment thus described; but there are many other ways of obtaining half a circle. Plane rectilineal figures are described under various heads; as trilateral or triangular; quadrilateral or quadrangular; and multilateral or polygonal. A triangle (fig. 8) is a plane rectilineal figure contained by three straight lines, which are called its sides. No figure can be formed of two straight lines; hence, an Fig. 8. angle is not a figure, its legs being unlimited as to length. Triangles are divided into various kinds, according to the relation of their sides or of their angles; as equilateral, isosceles, and scalene; right-angled, obtuse-angled, and acute-angled. Fig. 9. An equilateral (equal sided) triangle, Fig. 10. is that which has three equal straight lines or sides (fig. 9). An isosceles (equal-legged) triangle, is that which has only two equal sides (fig. 10). A scalene (unequal) triangle, is that which has all its sides unequal (fig. 11). Fig. 11. A right-angled triangle is that which has one of Dits angles a right angle (fig. 11), in which the angle at A is the right angle. The side opposite to the right angle is called the hypotenuse (the subtense, or line stretched under the right angle) and the other two sides are called the base and the perpendicular; the two latter being interchangeable according to the position of the triangle. When one straight line meets another, at any point between its extremities, and makes the adjacent angles unequal to each other, each of them is called an oblique angle; that which is Fig. 6. greater than a right angle is called an obtuse angle and that which is less than a right angle is called an acute angle. Thus, in fig. 6, the straight line AB meets the straight line CD, in the point A, and makes o the adjacent angles unequal to each other; each of these angles is therefore called an obtuse angle; the angle c A B, which is greater than a right angle, is called obtuse and the angle D A B, which is less than a right angle, is called acute. ע A plane figure in geometry, is a portion of a plane surface, inclosed by one or more lines, or boundaries. The sum of all the boundaries is called the perimeter of the figure, and the portion of surface contained within, is called its area. Fig. 7. A circle is a plane figure contained or bounded by a curve line, called the circumference or periphery, which is such that all straight lines drawn from a certain point within the figure to the circumference are equal to each other. This point is called the centre of the circle, and each of the straight lines is called the radius. The straight line drawn through the centre and terminated at both ends in the circumference, is called the diameter of the circle. It is plain, from the definition that all the radii must be equal to each other; that all the diameters must be equal to each other, and that the diameter is always double the radius. In speaking or writing, the circle is usually denoted by three letters placed, at any distance from each other, around the circumference; thus, in fig. 7, the circle is denoted by the letters A CB, or A E B; or, by any three of the other letters on the circumference. The point o is the centre, either of the L An obtuse-angled triangle is that which has one of its angles an obtuse angle (fig. 8). An acute-angled triangle is that which has all its angles acute; figs. 9 and 10 are examples as to the angles, but there is no restriction as to the sides. In any triangle, a straight line drawn from the vertex of one of its angles perpendicular to the opposite side, or to that side produced (that is extended beyond either of its extremites in a continued straight line), is called the perpendicular of the triangle; as in fig. 12, where the dotted line is the perpendicular of the triangle; and in fig. 13, where the dotted line drawn from the point a to the dotted part of the base produced is the perpendicular of the triangle. Fig. 12. Σ Fig. 13. F A quadrilateral figure, or quadrangle, is a plane rectilineal figure contained by four straight lines, called its sides. The straight line which joins the vertices of any two of its opposite angles, is called its diagonal. Quadrangles are divided into various kinds, according to the relation of their sides and angles; as parallelograms, including the rectangle, the square, the rhombus, and the rhom boid; and trapeziums, including the a Fig. 14. trapezoid. A parallelogram is a plane quadrilateral figure, whose opposite sides are parallel; thus, fig. 14, A C B D, is a parallelogram, and ▲ B, CD, are its diagonals. Fig. 15. A rectangle is a parallelogram, Fig. 16. A square is a rectangle, whose A rhomboid is a parallelogram, whose angles are oblique (fig. 14). A Fig. 17. A rhombus, or lozenge, is a rhomboid, whose sides are all equal (fig. 17). A trapezium is a plane B quadrilateral figure, whose Fig. 18. A trapezoid is a plane quadrilateral figure, which two of its sides parallel (fig. 19). C B Fig. 19. Words may be divided into two classes; first, those which are primitive: secondly, those which are derived. Primitives, or original words, are words which have been purposely formed to express one idea; they are words which have not been taken from any other word, and which cannot be reduced so as to be more simple. Thus, the words man, book, child, house, &c., express a complete sense, and nothing can be taken from them. Derivatives are words which are drawn out of others, or which take part or parts of others, or which are formed by joining two words together, so as to make a new word meaning something different from the others; as mankind, bookbinder, childlike, housekeeper, &c. All these words you will see may be reduced; mankind may be reduced to man, and still express a has complete sense; and so may the other words. There are ten sorts of words in the English language: these are commonly called PARTS of SPEECH, and the names given to them are as follow:-Article, Noun, Pronoun, Adjective, Verb, Participle, Adverb, Conjunction, Preposition, Interjection. Each of these must be studied separately. That you may have a general view of the whole subject, a brief mention of them will be made here, though each will afterwards be exguage there are about sixty thousand words; of course, each of these words belongs to one or other of the following ten parts of speech. 1. The ARTICLE is placed before a noun, to point it out, and to fix its exact meaning. A multilateral figure or polygon, is a plane rectilineal figure, of any number of sides. The term is generally applied to any figure whose sides exceed four in number. Polygons are divided into regular and irregular; the former having all their sides and angles equal to each other; and the latter having any variation what-plained more fully. It is supposed that in the English lanFig. 20. ever in these respects. The sum of all the sides of a polygon is called its perimeter, and when viewed in position its contour. Irregular polygons are also divided into convex and non-convex; or, those whose angles are all salient, and those of which one or more are re-entrant. The irregular polygon (fig. 20), has its angles at B, C, and D, salient; and its angles at A and E, re-entrant. Polygons are also divided into classes, according to the number of their sides; as, the penFig. 21. Fig. 22. tagon (fig. 21), having five sides; the hexagon (fig. 22), having six sides; the heptagon having seven sides; the octagon having eight sides; and so on, According to this nomenclature, the triangle is called a trigon, and the quadrangle a tetragon. QUESTIONS ON THE PRECEDING LESSON. What is a geometrical solid? what is a point? a line? a straight line? a crooked line? a curve? How are lines denoted? what is a surface? a plane? an angle ? What are parallel straight lines? how are angles denoted? how are they divided? what is a right angle? an oblique angle? an acute angle? an obtuse angle? What is a plane figure its perimeter? its area? a circle? its circumference? its centre? its radius? its diameter ? How is a circle denoted? what is an arc? a chord? a segment? a sector? a semicircle? a How are plane rectilineal figures divided? what is a triangle? an equilateral triangle ? an isosceles? a scalene? a right-angled triangle? an obtuse-angled? an acute-angled? the perpendicular of triangle? In a right-angled triangle, what are the hypotenuse, the base, and the perpendicular? What is a quadrilateral figure? how are quadrangles divided? what is a diagonal? a parallelogram? a rectangle? a square? a rhombus? a rhomboid? a trapezium? a trapezoid? a polygon? How are polygons divided? what is the difference between regular and irregular polygons? what is perimeter ? contour? salient? re-entrant? pentagon? hexagon? heptagon? trigon? tetragon? LESSONS IN ENGLISH GRAMMAR.-No. II. ETYMOLOGY. SUPPOSING that you have acquired a pretty good knowledge of the sounds of the different letters in the alphabet, you should next apply yourself to the study of the most correct way of putting them together, so as to form words rightly spelled, and sentences properly constructed. To do this you must find out how words have been formed-how many different sorts of words there are-what is their exact meaning-and the way in which they may be changed or altered, according to the ideas which the writer or speaker may wish to express. This is the second part of English Grammar, and it is called ETYMOLOGY, 2. The NOUN is the name of anything. 3. The PRONOUN is used for, or instead of a noun, to prevent its being repeated too often. 4. The ADJECTIVE expresses the particular quality or property of the noun. 5. The VERB expresses action, being, or suffering. 6. The PARTICIPLE is a word derived from a verb, and partakes of the nature both of a verb and an adjective. 7. The ADVERB describes the quality, or circumstance, or peculiar meaning of other words, and is joined either to a verb, an adjective, a participle, or another adverb. 8. The CONJUNCTION joins the several words or parts of sentences together. 9. The PREPOSITION is commonly set before words to connect them, or to show their relation. The INTERJECTION expresses some sudden emotion of mind. I. THE ARTicle. them out, and to fix their exact meaning. There are in the THE ARTICLE is a part of speech set before nouns, to point English language two articles, a or an, and the. The a and an are reckoned but as one, because becomes an when it is placed before a vowel; that is, before a, e, i, o, and u short, as an urn; also before a mute, or an h that is not sounded, as an hour; if the h be sounded, the article a only is used, as, a haven. The article a is used before all words beginning with a consonant, as, a shoe, a boot; also before u long, that is, when it has the sound of you, as in, a useful book, not an useful book, or, a union, not an union. Articles are divided into two classes, definite and indefinite. An indefinite article is one which does not define the particular meaning or application of the word before which it is placed; or which speaks of things in general, things which are com mon, or of which there are many of the same sort. A and an are indefinite for this reason; as if you were to say, a book, a man, an apple, you would not be understood as meaning any particular book, or man, or apple. The is called the definite article, because it defines your meaning, and fixes it to one particular thing; as, the book, the man, the apple; that is, some particular book, or man, or apple. If you were to enter a coffee-room, and say to the waiter, "Lend me a paper," he would bring you the first that came to hand; but if you saw the Times in his hand, and said, lend me the paper," he would understand you as asking for that particular paper. For this reason the definite article is sometimes called the demonstrative article, as, when pointing to an individual, we say, "That is the man I wish to employ;" or "This is the book I want you to lend me." The article the may be set before nouns both of the singular and plural number, because we can speak definitely of many as well as of one; as, the men, the books, the apples. |