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eonverting them into a new compound, whose properties adapt Having thus learned how bone is formed, and how it it to become part of the growing organised substance.
acquires increased strength and size, let us now see whether So in animal structure. So in the human body. We have we can find out the number of bones in the human body, how selected the bones with which to begin. All bone is developed they are held together, and what purpose they serve. Here is from a substance called CARTILAGE, which is a species of gristle. a skeleton which will help us to trace the connexion of the In its simpler form, this cartilage is composed entirely of cells, different parts of this wondrous structure, and show us how with a jelly-like substance between, called GELATINĚ. These beautifully fitted it is for the ends designed by the great cells are found in little clusters of two, three, and four; but as Creator. It includes one hundred and ninety-eight bones. they begin to take on the character of bone, instead of these These are exclusive of the teeth, of which there are sixteen smaller clusters, we have groups made up of a much larger inserted in each jaw, and also exclusive of a few small bones, number. These groups are still separated by the jelly-like which serve to lengthen and give additional strength to some substance which comes between, and it is in this substance others. We shall now arrange these one hundred and ninetythat the bony matter is first deposited. Gradually, assuming eight bones in the form of a table, that you may see more the form of deep narrow cups of bone, it receives the ends of clearly and remember more easily their connexion :the cartilage cells, as these become arranged in long rows.
In the cranium or head
8 Then, these cartilage cells become consolidated, and take on the
In the spine, called the vertebræ, including the hardness of bone. When the temporary cartilage is converted
sacrum and the coccyx, which are called false into bone, the bone has still to be enlarged in conformity with
vertebræ, because they adhere to each other and the increasing size of the surrounding parts. How is this
do not move ..................
26 enlargement insured : In two ways. All the new cartilage
In the face.........
14 at the edges and on the surface of the bone, becomes ossified,
A little bone which lies at the root of the tongue, and it is known that the growth of a long bone takes place
called Os-hyoides, of this shape 0..................
1 chiefly towards its extremities, while the bony matter on its
The ribs—and the breast-bone to which they are surface contributes to increase its thickness. This is one
in part attached
25 method. The other is by having a cavity in the bone itself.
In each superior extremity, including the shoulder, You know that in mechanics, when we want to get the greatest
arm, fore-arm, and hand 32
64 strength with a limited amount of material, we choose a hollow
In each lower extremity, including the pelvis, cylinder. Just so in the human body. The bones are designed
thigh, leg, and foot 30...........
60 for strength and support; and instead of being a solid mass, they most of them are hollow. The matter first deposited on
198 the inner surface of these tubes or hollow bones, is pushed
If we examine the composition of bone, we shall find that it Parietal. Frontal.
contains the three essential elements, hydrogen, nitrogen, and oxygen ; with a certain proportion of lime and phosphorus,
or phosphate of lime. It is by the excess of this earthy Temporal.
matter in the bones of old people, that they are so easily and
so frequently broken. We seldom if ever hear of a child inferior Maxillary.
breaking a bone, and yet how often do children fall, how sadly
are they sometimes bruised. Cervical Vertebrae.
The purpose of bone in the animal economy is purely mechaScapula.
nical. "Not only do the bones serve as points of attachment to the muscles in the midst of which they are situated, but afford support and protection to the softer textures, and form inflexi. ble levers, on which the muscles may act, and give motion to the different parts of the great fabric. The brain is a soft and
delicate texture, and to preserve which uninjured, is essential II umerus.
to the play and activity of the mind. Now see how admirably Luinbar
this is insured by incasing it within the hard and bony subVertebræ.
stance of the cranium. So the heart and the lungs are
defended by the ribs, and the spinal marrow by the vertebræ Lium
which compose the back bone. In all this, we behold the wise
and benevolent arrangement of the Creator. If the brain had Ulna
not been so defended, it would have been liable to constant
injury, and in the proportion of the injury must the intellect Radius
have been impaired. Carpus
Or, suppose we break a bone, how disabled do we become? Metacarpus If it be in the leg, we have no longer the power of locomotion,
or of moving from one place to another. If it be in the hand, Fingers
we are no longer qualified to perform the every day duties of life. But here again we see the goodness of God. While the bones are subject to manifold diseases and external injuries,
there is no other structure of the same complex nature, which Rotula
is capable of being so thoroughly repaired. There have been
cases of the re-formation of nearly an entire bone, when the Filula
original one had been lost by disease. So in a fracture. Any portion of the shattered bone that remains connected with the
surrounding membrane, and the vessels with which it is supTibia
plied, becomes the centre of a new formation, and the injury is repaired in a few weeks.
You know that the bones exhibit a great many joints, and that it is by these joints we are enabled to move our bodies in such a variety of ways. But for such an arrangement in the fingers, for example, and we could neither close our hands, nor grasp any object on which we wished to lay hold. Now if there be so many joints, how is it that the
bones are preserved each in its place? This is done by outwards by succeeding layers, and thus the bone gradually ligaments and cartilages. Cartilage is a white elastic substance acquires an increased diameter,
next to bone in solidity; ligament is a strong, whitish, flexible,
fibrous substance; and both in some instances supply the In studying the ensuing lessons, you must implicitiy follow place of bone ; in other instances they fix the
ends of the bones my directions. I have been for many years engaged in teaching. so as to confine the motions of the joint; sometimes they give and, from my experience, know that there is no obstacle to origin to muscles, and sometimes they fix the bones almost progress greater than that which scholars create for themselves, immovably together. . Between the joints there is a fuid in giving preference to their own judgments and following called synovia, which is of a yellowish hue, like olive oil, and their own fancies and opinions. In your practice, acknowledge which like oil, serves to lubricate and render easy the motion and observe it as a first principle, that your instructor knows of one bone upon another.
better than you. Take it for certain that he is right, until How beautiful and how gracioris are these arrangements ! should it so happen—you have proved that he is wrong. At How worthy of that wise and loving Creator, the purpose and the same time, scrupulously follow my directions. Do not end of all whose works is not less our happiness than his own attempt to get before me; take care not to fall behind me. glory! With what interest should we study His works, and Do what I bid, do it when I bid it, dą it as I bid it. The more how warm and grateful should be the praise of our hearts ! rigid you are with yourself, in obeying these injunctions, the But we have gone far enough for one lesson. Let us now
more certain and the more rapid will your progress be. As put the whole in the form of questions, and see how much you teach you, and as having for more than a quarter of a century
having myself gone through every thing that I am about to have learned, and how much you remember.
been engaged in teaching these things to others, I know What are the three kingdoms into which the world of nature what difficulties are in the way, and I have learned how to is divided ?
diminish or remove them. Is the distinction between living bodies and dead matter to be confidence; and if you are not willing to give me your con
On these grounds I claim your always easily traced ? Give an example of this difficulty.
fidence, you had better not enter on the study of the Latin With which of these three kingdoms has human physiology language. to do?
In the instructions which I am to give you, I shall suppose Of what material is the human body made up ?
myself addressing a friend, who besides some general acIs there any difference between the particles of which the body quaintance with his mother tongue, has acquired from the is made up and the atoms of the earth's surface ?
English Lessons in the POPULAR EDUCATOR, or from some What do you mean by organised matter?
other source, a knowledge of the ordinary terms of English What are the essential elements in all organic or animal sub- Grammar, such as singular, plural, noun, adjective, verb, stance ?
adverb, &c. The meaning of such words I shall not explain. Does chemical combination produce life? What do you mean by human physiology?
But everything peculiar as between the English and the Latir. From what does all animal substance spring ?
I shall explain. I shall also explain any grammatical term, Of what is the germ formed ?
which though used sometimes in English Grammar, you pos. Whence does it get materials for its nourishment and growth? sibly may not understand. In my explanations I think it Give an example from the vegetable kingdom.
safer to err on the side of superfluity rather than on the side From what substance is bone developed ?
of deficiency. I have said that I shall suppose you to possess How is cartilage formed ?
a general acquaintance with the English language. But I In what substance does the bony matter first begin to form ?
advise you to suspect yourself as being probably acquainted How is the growth of bone insured ?
with it, but in a very imperfect manner. And this advice I How many bones are there in the human skeleton ?
give you in the hope that it may lead you to the constant use Give the number in each part.
of a good English Dictionary. In every case in which you How many teeth are there in each jaw ?
have the least doubt whether or not you know the exact What purpose do the bones serve Give an example of the protection which they afford to some of meaning of any word I use, look out the word in your dic. the softer textures.
tionary, and put it down in a note-book to be kept for the When a bone has suffered from disease, is it easily and ever purpose. Having, written it in the note-book, add the meanthoroughly re-formed?
ing. When you have, say a score of words thus entered in What holds the bones together ?
your note book, look them over again and again until their What substance is found between the joints to make the bones signification is impressed on your memory. If you listen to move more easily the one upon the other?
this suggestion, and continue to make progress with me, you will soon find numerous exemplifications of the assertion 1 made but now, namely, that a large proportion of the worda
in the English language are of Latin origin. Take, for inLESSONS IN LATIN.—No. I.
stance, the last sentence. In that sentence alone the following By John R. BEARD, D.D.
words are derived from the Latin : I mean suggestion, continua, INTRODUCTORY.
progress, numerous, exemplification, assertion, proportion, language.
Latin, origin of the two-and-forty words of which the senBeing about to give you, reader, some lessons which may enable tence consists, ten are from the Latin. Should you ever you to learn the Latin language, with no other resources possess an acquaintance with the science of philology, or the than such as may be supplied by
your own care and diligence, science of languages, you will know that in the sentence there I take it for granted that you are desirous of acquiring the are other words which are found in the Latin as well as in necessary skill, and am willing to bestow the necessary labour. other ancient languages. Independently of this, you now If the study were not recommended as a good mental disci- learn that about one-fourth of our English words have come pline; if it were not recommended as giving a key to some of to us from the people who spoke Latin, that is the Romans the finest treasures of literature ; if it were not recommended and other nations of Italy. In reality, the proportion of Latin as a means of leading you into communion with such minds as words in the English is much greater, as in time you may those of Cicero, Virgil, Horace, Livy, and Tacitus, it would have know. Observe, too, that these Latin words in the sentence a sufficient claim on your attention, as greatly conducing to a are the long and the hard words, are what perhaps you may full and accurate aquaintance with your mother tongue—the call “ Dictionary words." These are the very words which English. The English language is, for the most part, made up give you trouble when you read an English classic, or firstof two elements : the Saxon element, the Latin element. rate author. But they give me no trouble. With me, they Without a knowledge of both these elements, you cannot be are as easy to be understood as any common Saxon term, such said to know English. If you are familiar with both these as father, house, tree. The reason why they have long ceased elements, you possess means of knowing and writing English, to give me trouble, is, that I am familiar with their roots, or the superior to the
means which are possessed by many who have elements of which they each consist. Having this familiarity, received what is called a classical education, and have spent I have no occasion to consult the dictionary. There are thouyears in learned universities. In order to be in possession of sands of English words of Latin origin, the meaning of which both those elements, you must, for the Saxon element, study I know, though I have never looked them out in a dictionary, German, and for the Latin element, study the lessons which I wish to assist you in putting yourself into a similar position; ensue,
and although you may have no aid but sucli as these pages
afford you, I do not despair of success, if only you will strictly other way of marking a long vowel, except by throwing on observe my requirements.
it, the accent or stress of the voice. It is also a fact, that the I shall, however, have little chance of carrying my wishes same vowel is sometimes short and sometimes long; in other into effect, if you begin with a notion that the task is an easy words, the same vowel sometimes has, and sometimes has not one. Were it an easy task, it would not be worth your trouble. the accent on it; thus the i in dominus, a lord, is without the I tell you plainly that the task is hard ; that it is a long task; accent, while the i in doctrina, learning, has the accent; the that it will require on your part patience and diligence. If former, therefore is pronounced thus, dóminus, the latter thus, you commence the study under the delusion that it is mere doctrina. Now observe that these words are trisyllables, as play, you will do as very many have done before you,--in a dom, i, nus. Of these three syllables the last, namely us, is short time you will grow weary, and give it up. But if, in a called the ultimate; the second, i, is called the penult; the true manly spuit, you take up the task as a worthy task, as a third, or dom, is called the antepenult. And the general rule useful though a difficult task, as a task in which whatever I for pronouncing Latin words is, that the accent is thrown on may do for you, you must do far more for yourself, then you the penult, or if not on the penult, then on the antepenult. will not fail to make progress ; your course will become easier In doctrina the accent is on the penult, or last syllable but as you go on; and you will have the delightful satisfaction one. In dóminus, the accent is on the antepenult, or last that you are employing your time and your faculties so as to syllable but two. In order that you may know where to lay produce definite, important, and abiding results. If, however, the stress of your voice. I shall mark, as in dóminus and it is not an easy thing to acquire an acquaintance with the doctrina, on which syllable the accent lies. You will then principles of the Latin language, then you will at once see understand that when I put a mark thus ' over a vowel, I that it is not wise to be engaged at the same time in other mean thereby that you should let your voice rest, as it were difficult studies. Many persons, especially young persons, on that vowel. For example, in the word incur, the accent fail to give themselves a good education, because they attempt you know is on the last syllable, for you throw the stress of too much--because, indeed, they attempt what is impossible. the voice on the syllable cur. This is indicated thus, incúr. I remember that when I began in earnest to study Latin, I So in the Latin amicus, a friend, the accent is on the i, was saved from this danger, by the judicious opinion of a and the word is to be pronounced thus amicus, the accent learned and venerated friend, who advised me not to attempt being on the penult. There is another way of marking the more than one subject at a time. In part, my friend, I am same fact; it is by the use of a short strait line, as ", and a like you, self-taught. In the ordinary phrase, indeed, I learnt curve, as o. The former denotes a long or accented syllable, Latin when a boy at school ; but so wretched were the methods for instance doctrina; the latter denotes a short or unaccented of teaching some forty years ago, that though I learnt Latin syllable, for instance, dominus. You thus see that doctrina I knew nothing of Latin when I left school. Soon, however, and doctrina, dóminus and dominus point out the same thing, 1 began the study in earnest; I began almost unaided. But namely, that in pronouncing doctrina you must lay the stress an adviser came, and under his directions I for a time confined of the voice on the i, and in pronouncing dóminus you must myself almost exclusively to my Latin studies. They were lay it on the o. hard, very hard ; much harder were they to me than they need I must point out to you another practice. In Latin, as you be to you. But by giving my mind perseveringly to the task, will presently learn, the endings of words have a good deal I reaped success, and so in time became prepared for college. to do with their meanings. It is, on that account, usual to Let me be your adviser. Even in the treasures of the Popular pronounce them at least very distinctly. Indeed, I might say, EDUCATOR you may find a source of danger. If you engage in that on every terminating syllable a sort of secondary accent is all the studies here offered, you will become proficient in none. laid. Thus, dominus is pronounced dóminûs. So in other Make a selection. If you wish to learn Latin, pursue the forms of the word, thus, dóminí, dóminó, dóminum. The object is study wisely. But wisely you cannot pursue it, if you com- to mark the distinction between, say, dominus and domino, a bine therewith several other subjects. For you, one language distinction of great consequence. Another form of this word at a time is enough. Do not attempt more. If you want is dominos. For the same reason a stress is laid on the termirelief from this, which is a severe study, unite with it the nation os, which accordingly is pronounced as if it were oase. Lessons in History which these pages supply, But do not Words, too, which end in es have a secondary accent on attempt to learn French while you are learning Latin. When the e; as Vulpes, a jox, pronounced vulpees. In a few cases you know Latin, you will find French very easy. With a the vowel is what we call doubtful, that is, it is sometimes knowledge of Latin you would also have no difficulty at all in short and sometimes long. This peculiarity is marked thus, becoming, in a short time, acquainted with the Italian, the Spanish, and the Portuguese languages, which have Latin for penult, as tenebrae, or on the antepenult
, as ténebrae. Observe,
as in tenebrae, darkness, when the accent may be on the their common parent. In the Latin, too, you would find a great assistance should you wish to learn Greek, and so acquire also, that a vowel at the end of a word is always pronounced the power of reading the scriptures of the New Testament in in Latin. Take, as an example, docéré, to teach, which is their original tongue,
pronounced as it is marked, that is, with an accent on the last
syllable no less than on the last syllable but one. The Latin You may practically regard the Latin alphabet as the same as the English. The English letters may be traced to the language has no silent e, as we have: for instance, in wife. Latin; the Latin letters are derived from the Greek; and the Practise yourself, according to these rules, in pronouncing thu3 Greek are in substance identical with the Phenician;
while the opening lines of that fine poem, Virgil's Aeneid. As I am the Phenician alphabet is the oldest, or one of the oldest, in its meaning, 1 subjoin the translation made by the English
anxious that you should not pass anything without knowing the world,
In the pronunciation, too, you may in the main follow the poet Dryden.
Inférrétque Deos Latió; genus únde Latinum,
Albánique patrés, á' que áltaé móonia Rómae." Consequently the pronunciation is not a matter of conse
“ Arms and the man I sing, who forc'd by fate, quence. Even in our own country there are diversities.
And haughty Juno's unrelenting hate, but such diversities are secondary matters. To one or two
Expellid and exil'd, left the Trojan shore, remarks, however, you should carefully attend. In Latin the
Long labours, both by sea and land, he bore, vowels are what is called long or short. In other words, on
And in the doubtful war, before he won
The Latin realm, and built ihe destined town-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
His banish'd gods restor'd to rites divine,
And settled sure succession in his line. vowel i is mostly short. A long vowel is said to be equal to From whence the race of Alban fathers come, *wo short vowels. We English people, however, have no And the long glories of majestic Rome,'
In pronouncing the third line, you must cut off the um
EXERCISE-ENGLISH-LATIN. before the vowel i ; 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 the e 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. che vowel a in altae ; pronounced as if written qualtae. Accu
REMARK.-In order to make my meaning quite clear, I will racy of pronunciation, however, is not easily acquired from any
From cursus comes the written or printed directions. The living tongue is the only myself do this exercise in part. adequate teacher. And it will be well if you can get some
English word course; from in and curs comes incursion; from grammar school-boy to read to you and hear you read the 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 passage I have given above from Virgil, and the exercises, or some of them, which ensue. Although the pronunciation of out French words corresponding to, and derived from, Latin Latin is of secondary importance, yet you must try to be as words; as in courir, French to run; cours,
By correct as you can, if only from the consideration that what is comparison he may occasionally find that the same sound or worth doing at all, is worth doing well. But should you, as
word has a different meaning in French from what it has in you justifiably may, hope by these lessons to prepare yourself Latin or in English. Thus, concursus in Latin means a for becoming even a teacher of Latin-say in a school--you coming together, as to a meeting, a concourse of people; but
the would in that capacity find the pronunciation considered as a
corresponding French, concours, signifies co-operation. So matter of consequence ; indeed a disproportionate value is, concurrence in English is agreement, but in French competition. especially in the old grammar schools, attached to the esta- By practising comparisons such as this, you will not only meet blished methods of pronunciation. After all
, we cannot pro- with many curious facts, but be assisted to understand the nounce 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 or curs should bear dissimilar meanings, a little reflection
cipline, If it seems strange to you that the same letters curr 80 as to reproduce its music. In regard to the exercises which I am about to give, you curr. Its primary meaning is to run. Now, men may run
will take away your surprise. Go to the primary meaning of mechanical trade, you may repeat the words over again and into, or run out of, or run together, or run about, for different again while engaged in labour. Or you may make the words purposes.. For instance, they may run together in harmony, your own while walking to and from your employment. and then they concur; or they may run together in rivalry, Among my personal friends, is a gentleman who acquired the and then they are in what the French call concurrence, that is
, greater part of the words of the French language, while rising
competition. and dressing in the morning. Thousands of words have I 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 myself learnt while walking for recreation.
Having thoroughly mastered the vocabulary, take a slate if you follow these intimations, you will in time become not and write down the Latin into English ; then write the only a Latin scholar, but a good linguist. 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
LESSONS IN GEOMETRY.-No. I. Latin, and the Latin will assist you in writing the English. When you have got both the Latin and the English into as The term Geometry, which comes from the Greek word Geome. correct a state as you can, copy them neatly into a note-book. tria, literally signifies land-measuring, and was originally applied Having done so, read them carefully over, and compare each to the practical purpose which its name signifies, in the land instance with the rule or the direction, and also the example. of Egypt, the cradle of the arts and sciences. Herodotus, the Leave nothing until you understand the reason.
All the ex. oldest historian, with the exception of Moses, whose works amples or illustrations that I give, as well as the chief rules, have reached us, gives the following account of its origin : “I should be committed to memory. Before you proceed to a
was informed by the priests at Thebes, that king Sesostris second lesson, ascertain that you are master of the first. It made a distribution of the territory of Egypt among all his would be useful to write out the rules in one consecutive view, subjects, assigning to each an equal portion of land, in the in order that, having them all at once under your eye, you form of a quadrangle, and that from these allotments he used may study them in their connexion and as a whole, so as to see to derive his revenue, by exacting every year a certain tax. their bearing one upon another, and the general results to In cases, however, where a part of the land was washed away which they lead. Such a practice would have a very beneficial by the annual inundations of the Nile, the proprietor was effect on your mind, by habituating it to arrangement and permitted to present himself before the king, and signify what order, and might be expected to afford you valuable aid, both
in had 'appened. The king then used to send proper officers to other studies and in your business pursuits. Carefully avoid ex mine and ascertain, by admeasurement, how much of the haste and slovenliness. Do your best in all that you under- land had been washed away, in order that the amount of the take. “Well” not “much" should be your watch word. tax to be paid for the future might be proportional to the land Repeated reviews of the ground passed over are very desirable. which remained. From this circumstance, I am of opinion, Every Saturday you should go carefully over what you have that geoinetry derived its origin ; and from hence it was transdone during the week. At the end of every month the work mitted into Greece.” The existence of the pyramids, the of the month should be reviewed. On arriving at a natural ruins of the temples, and the other architectural remains of division of our subject, as for instance, when we have treated ancient Egypt, supply evidence that they possessed sone of the nouns, you should go over, and put together in your knowledge of geometry, even in the higher sense in which we mind the substance of what has been said thereon. “Be not now use the term ; although it is possiple that the geometrical weary in well doing, for in due season you will reap, if you properties of figures, necessary for the construction of such faint not. (Gal, vi. 9).
works, might have been known only in the form of practical
rules, without any scientific arrangement of geometrical truths, VOCABULARY.
such as are presented to us in the Elements of Euclid. Curro, I run. The chief parts are curro, Irun; currere, to run i meaning, signifies the science of space; or that science which
The word geometry, used in its highest and most extensive or the element in English derived from the parts, is curr; investigates and treats of the properties of, and relations existalso curs or cours. Con, from cum, means with; dis signifies in ing among, definite portions of space, under the abstract different directions; ex signifies out of.
division of lines, angles, surfaces, and volumes, without any
regard to the physical properties of the bodies to which they EXERCISE--LATIN-ENGLISK.
belong. In this sense, it appears to be very doubtful whether Curro and its parts give rise to several English words, as current, the Egyptians or Chaldeans knew anything of the science. It ("the current coin of the realm;") currency (" the circulating is to the Greeks, therefore, that we must look for the real medium.") Another example is found in the phrase "account origin of geoinetry, as an abstract science. Thales, the Greek eurrent."
philosopher, born 640 B.C., is reported, by ancient historians
to have astonished even the Egyptians by his knowledge of subject, and others connected with it, we cannot now enter ; this science. The founder of scientific geometry in Greece, but we trust to be able, in future numbers of this work, to however, appears to have been Pythagoras, who was born bring before our readers both its history and its application, about 568 B.C. He discovered the celebrated 47th proposition as one of the greatest of our modern engines in the discovery of the first book of Euclid's Elements, and various other of scientific truth, and in the development of the philosophy valuable and important theorems. He was great also in of nature. astronomy, having anticipated the Copernican system of the In giving our frst lessons on geometry, we think it advisable world. Of Plato, another great geometrician, and founder of to follow what seems to have been the natural course of events the academy at Athens, we have already spoken in our first in the history of this science. The present advanced state of number. He was the first who made some advances into what our geometrical knowledge was preceded in early times by a is called the higher geometry. The next name, super-eminent species of practical geometry gathered from experience, and in the science of geometry, is that of Euclid, whose Elements suited to the wants of those who required its application, behave been the prencipal text-book for learners, during a period of fore any attempt was made to enter very deeply into the study more than 2000 years. He flourished at Alexandria, in Egypt, of the theory. The latter was left to the schools of the about B.C. 300, during the reign of Ptolemy Lagus, who was philosophers and the academy of Plato. Accordingly, we one of his pupils, and to whom he made the celebrated reply, shall precede our disquisitions on the Elements of Euclid and when asked if there was a shorter way to geometry than by other geometers, both ancient and modern, by a short system studying his Elements :—"No, sire, there is no royal road to of practical rules and easy explanations in this important geometry.”
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.O. 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
DEFINITIONS. part of the ancient Greek geometry. He was the first who Extension, or the space which anybody in nature occupies, attempted to solve the celebrated problem of the rectification has three dimensions, viz., length, breadth, and thickness. of the circle; that is, finding a straight line exactly equal to the This is Euclid's definition of a geometrıcal solid. circumference. He found out the beautiful ratios of the A point is the beginning of extension, but no part of it; cylinder to its inscribed sphere and cone, and the quadrature hence it is said to have position in space, but no magnitude. of one of the conic sections. We have, in our second lesson A line is extension in one direction only; hence, it is said to on arithmetic, alluded to his discoveries in that science. His have length without breadth. Hence, also, the extremities of discoveries in physics or natural philosophy are simple, true, a line are points; and lines intersect or cross each other only and beautiful. The story of the determination of the specific in points. gravity of the golden crown of his cousin, Hiero king of A straight line is said, by Euclid, to be that which lies evenly Syracuse, is well known; and the very natural shout of between its extreme points; and, by Archimedes, to be the “ Heureka, heureka!"-I have found it, I have found it! on shortest distance between any two points. Both of these defini. coming out of the bath, has become a "household word.”' tions are defective; the defect is supplied thus : A straight Scarcely less celebrated was the famous Apollonius of Perga, line is such, that if any two points be taken in it, the part in Pamphylia, who flourished at Alexandria in the reign of which they intercept (or which lies between them), is the Ptolemy Euergetes, (from B.C. 247 to 222) another king of the shortest line that can be drawn between those points. same Ptolomean dynasty, and who was called by his cotempo- A crooked line is one composed of straight lines joined at raries the “Great Geometer." He wrote several books, full their extremities in any manner whatever, except that of of discoveries, on the higher geometry, and greatly extended uniform direction. A curved line, or curve, is a line whose the domains of the plane geometry. Other geometricians of direction varies at every point. eminence arose in the school of Alexandria, and bequeathed Straight lines, or curve lines, are generally denoted, in the precious remains of their genius to happier times. Clau-speaking and writing, by two letters placed commonly at their dius Ptolemæus, the author of the great work on astronomy extremities; but, they may be
Fig. 1. called Megale Syntaxis, the Great Construction, or Almagest; placed anywhere on the lines Pappus, the author of the Mathematical Collections : and others, at a distance from each other. including Theon and his daughter Hypatia, bring us down to Thus, in fig. 1, the letters A, B, the period when the Alexandrian library was burnt by com- denote one straight line; the mand of the Mohammedan barbarian Caliph Omar, and the letters C, D, another; and the labour and learning of ages were irrevocably destroyed. The letters E, F, a third; and these dark ages supervened, and little was done in the advancement straight lines are respectively of science until the glorious invention of printing, and the called the straight lines A B, general revival of literature about the middle of the fifteenth C D, and E P. A straight line, century.
as A B, may be divided into The ancient Greek geometry was speedily made known to any number of equal parts, to serve as a standard for the moderns through the medium of translations of, and com- measuring other straight lines. mentaries upon, the writings of the great masters. The Ele- A comination of straight, crooked, and curved lines is re. ments of Euclid, indeed, were reckoned so perfect, that no
presented in fig. 2; A B, BC, D, and Da, attempt was made to supersede them; and the only object of
are each straight lines; the combination writers on geometry in general was to explain his works, and
ADC B, beginning at A, and terminating to make what additions they could to the science, in the same
at B, is a crooked line; and the line A MB, masterly style of composition. A host of names of eminent
beginning at a, and ending at B, is a authors might be mentioned, who succeeded in establishing
curved line. the Greek geometry, and in extending its domains. The prin D
C A surface, or, as it is sometimes called, cipal of these, however, was Dr. Robert Simson, professor of a superficios, is extension in two directions; hence, it is said to mathematics in the University of Glasgow, who flourished in have only length and breadth. Hence, also, the extremities the middle of the last century. His grand endeavour was to or boundaries of a surface are lines; and surfaces intersect or present to modern Europe the Elements of Euclid, as they cross each other in lines. originally appeared in ancient Greece. In this he succeeded A plane surface, or plane, is a surface in which any two points to admiration, and his edition of this great work maintains its being taken, the straight line between them lies wholly in that reputation to the present moment. He was also an original surface; or, it is that surface with which a straight line wbolly writer of great eminence; and but for the eclat of the new coincides, when applied to it in every direction. Any other geometry, invented by Leibnitz and Newtor, he would have surface, not composed of plane surfaces, is called a curved shong as a star of the first magnitude. On this interesting surface.