printed poems, entitled "Early Blossoms," were issued in 1808, when she was fourteen. In 1809 her family removed from Gwrych to Bronwylfa, near St. Asaph, in Flintshire, where she resided for sixteen years and wrote many of her works. It was during this year, 1809, that the great event of her life took place--her introduction to Captain Hemans. The young poetess was then only fifteen, in the full glow of that beauty which was destined to fade so early. The mantling bioom of her cheeks was shaded by a profusion of natural ringlets, of a rich golden brown, and the ever-varying expression of her brilliant eyes gave a changeful play to her countenance, which would have made it impossible for any painter to do justice to it. No wonder that so fair a creature should excite the admiration of a gallant captain. And the love on both sides was ardent and sincere, for Captain Hemans, soon after their introduction, was called upon to embark with his regiment for Spain. On his return in 1812, they were married. Mrs. Hemans' eagerness for knowledge continued to be intense, and of her industry volumes, still existing, of extracts and transcriptions are evidence. The mode of her studies was very desultory to outward appearance, as she loved to be surrounded by books of all sorts and languages and on every variety of topic, turning from one to another. And this course, it is said, "she pursued at all times-in season and out of season-by night and day-on her chair, her sofa and bedat home and abroad-invalid, convalescent and in perfect health-in rambles, journeys and visits-in company with her husband, and when her children were around her at hours usually devoted to domestic claims, as well as in the solitude of the study and bower." In the year 1818, Captain Hemans' health requiring the benefit of a warmer climate, he determined upon repairing to the Continent, and eventually fixed his residence at Rome. At this time a permanent separation was not contemplated by either party, and it was only a tacit and conventional arrangement, with a frequent interchange of correspondence relative to the education and disposal of the children. But years rolled on, and from that time till the hour of her death, Captain and Mrs. Hemans never met again. She continued to reside with her mother at Bronwylfa, and had the five boys left under her care; a sufficient proof that nothing more than incompatability of pursuits and uncongeniality of temper were the moving causes of their separation. Notwithstanding the peculiarity of her situation in consequence of this separation, her talents, her amiable qualities, and the increasing popularity of her writings, continued to secure to Mrs. Hemans the warm attachment of several distinguished friends, among whom were Bishop Luxmore and Bishop Heber. With the latter she became acquainted in 1820, and he was the first literary character with whom she ever familiarly associated. To him she submitted the commencement of a poem entitled "Superstition and Revelation," which was, however, never completed by her; and at his suggestion she was first led to offer her " Vespers of Palermo" to the stage. This play, completed in June, 1821, was, after many theatrical delays, acted at Covent Garden, in December, 1823, but proved a failure. It, however, led to a correspondence with the poet Milman, who kindly interested himself in its behalf, and it was subsequently acted at Edinburgh with considerable success, with an epilogue written by Sir Walter Scott. The death of her beloved mother, which occurred in 1827, was an irreparable loss to Mrs. Hemans; she had now no one to whom she could cling for protection, and her sensitive, dependent nature made the maternal shelter and security necessary to her happiness-almost to her own existence. As the care and education of her five sons now devolved entirely upon herself, she was induced to leave Wales, where her heart still clung, and settle at Wavertree, a small village near Liverpool, where she hoped to find superior advantages of education for her boys. During the many years that Mrs. Hemans resided with her mother, the anxieties and responsibilities of housekeeping had never fallen to her lot, and her time and thoughts might be and were almost exclusively devoted to poetry and literature. But now domestic cares forced themselves upon her attention, and butchers' and grocers' bills intruded, as she observes, "in frightful array." In these household duties she felt but little interest, being, as she playfully describes herself, "little better than a grown-up Rosamond [Miss Edgeworth's naughty girl], who constantly lie in bed till it is too late to get up earlybreak my needles (when I use any)--leave my keys among my necklaces-answer all my amusing letters first, and leave the others to their fate." Elsewhere she says, "I am now for the first time in my life holding the reins of government independently-managing a household myself, and I never liked anything less than ce triste empire de moi-même." In the summer of 1829 she visited Scotland, where she was cordially received by many distinguished visitors, among others, by Sir Walter Scott, with whom she spent two or three weeks very delightfully. When bidding her farewell, he said, "There are some whom we meet and should like ever after to claim as kith and kin, and you are one of these." On one occasion he observed: "One would say you had too many accomplishments, Mrs. Hemans, were they not all made to give pleasure to those around you." In 1830 Mrs. Hemans visited the Lakes, where she formed a personal acquaintance with Wordsworth, whose writings she had always admired. Mrs. Hemans was delighted with the scenery at Rydal Mount, and determined to hire a residence called Dove's Nest, beautifully situated in a very romantic spot on the banks of Windermere. She thus describes it in one of her letters :"The house was originally meant for a small villa, though it has long passed into the hands of farmers, and there is in consequence an air of neglect about the little demesne, which does not at all approach to desolation, and yet gives it something of touching interest. You see everywhere traces of love and care beginning to be effaced,--rose-trees spreading into wildness-laurels darkening the windows with their luxuriant branches; and I cannot help saying to myself, 'Perhaps some heart like my own in its feelings and sufferings, has here sought refuge and found repose.' The ground is laid out in rather an antiquated style, which, now that nature is beginning to reclaim it from art, I do not at all dislike. There is a little grassy terrace immediately under the window, descending to a small court with a circular grass-plot, in which grows one tall white rose-tree. You cannot imagine how I delight in that fair, solitary, neglected-looking tree. I am writing to you from an old-fashioned alcove in the little garden, round which the sweet-briar or the moss-rose tree had completely run wild, and I look down from it upon lovely Windermere, which seems at this moment even like another sky, so truly is every summer cloud and tint of azure pictured in its transparent mirror." In 1831 she left England with her children, to take up her residence permanently in Dublin. The next four years were passed busily and rather pleasantly by Mrs. Hemans, who continued to write unceasingly, though a gradual decline in her health was perceptible to her friends. At the close of the year 1834 her health became very precarious, and the following spring brought symptoms of her approaching dissolution. The closing scene has been impressively described by one of her friends: "Mrs. Hemans was now too ill to leave her room, and was only laid upon a couch during the daytime, occasionally suffering severely. But all was borne with resignation and patience; and when not able to bear even the fatigue of reading, she had recourse to her mental resources, and as she lay on the sofa, she would repeat to herself whole chapters of the Bible, and page after page of Milton and Wordsworth. Her thoughts reverted frequently to the days of her childhood -to the old house by the sea-shore-the mountain ramblesthe haunts and the books which had formed the delight of her childhood. She was wont to say to those who expressed pity for her situation, that she lived in a fair and happy world of her own, among gentle thoughts and pleasant images;' and in her intervals of pain she would observe, that 'no poetry could express, nor imagination conceive, the visions of blessedness that flitted across her fancy, and made her waking hours more delightful than those even that were given to temporary repose.' Indeed, her sister observes," At times her spirit would appear to be already half-etherealized, her mind would seem to be fraught with deep and holy and incommunicable thoughts, and she would entreat to be left perfectly alone in stillness and darkness, to commune with her own heart,' and reflect on the mercies of her Saviour." 716 I. Quartzy conglomerates and sandstone. On the 15th of March, after receiving the holy sacrament, from 8,000 to 10,000 feet, and are generally divided into three she became extremely ill, but a temporary improvement took groups :place, and on the 26th of April she dictated to her brother (for she had for some time been constrained to employ ar. amanuensis) her "Sabbath Sonnet," the last strain of the sweet singer of the hearth, the home and the affections. On Saturday, the 26th of May, she sank into a peaceful slumber, which continued all day, and at nine o'clock in the evening her gentle spirit passed away without pain or struggle. Her remains were deposited in a vault beneath St. Anne's Church, Dublin, almost close to the house where she died. A small tablet has been placed above the spot where she is laid, inscribed with her name, her age, and the date of her death, and with the following lines from a dirge of her own:"Calm on the bosom of thy God. Fair spirit! rest thee now! Dust to the narrow home beneath, They that have seen thy look in death, LESSONS IN GEOLOGY.-No. LXI. ON THE CLASSIFICATION OF ROCKS. SECTION XII. THE OLD RED SANDSTONE, OR THE DEVONIAN You have already been informed that there exists, above the There exists also another series of red sandstones, which generally lie beneath the mountain limestone, that is, beneath the foundation of the coal-measures. As this group of red sandstones had been formed long before the coal-measures, and consequently myriads of ages before the Trias, and as they contain only palæozoic fossils, they were called the Old Red Sandstone. At the present day the group of Old Red Sandstones are called the DEVONIAN Formation, their identity having been established by Sir RODERICK MURCHISON. These rocks were first studied in their grand developments in Herefordshire and Breconshire. When rocks of the same colour and mineral character were examined in Devonshire, the strata there were found to contain the shells of the genus Orthis, Spirifera, Productus, and Terebratula, which were not found in the rocks of Breconshire and Herefordshire, which again, on the other hand, afforded fossil fishes that were not to be found in Devonshire. But, in 1840, Sir R. Murchison visited Russia, and discovered in strata of the Old Red Sandstone, the fossil fishes contained in the rocks of Herefordshire and Scotland, associated with the same fossil shells that were furnished by the rocks of Devonshire. The same geologist, in connection with Professor SEDGWICK, have also shown, that in the Rhenish Provinces of Germany there exist beds equivalent to the Devonshire strata, and which contain both the corals and the shells which are characteristic of the Devonian rocks. It is not many years ago since the Old Red Sandstone was regarded as nothing but a lower division of the coal formation; but the labours of Murchison, Sedgwick, and Hugh Miller have shown, that it contained paleontological character of sufficient distinctness to mark it out as an independent group. The Devonian rocks are of enormous thickness, varying I. QUARTZY CONGLOMERATES AND SANDSTONE. 1. The uppermost beds, which consist of quartzy pebbles and grits, hardened together by a cement that is slightly limy or calcareous. These conglomerates are, in colour, pink or reddish. The quartzy pebbles are water-worn or rounded fragments of some of the older rocks, such as Silurian slates, gneiss, etc. In South Wales, these conglomerates are seen to great advantage at Symond's Yat, between Monmouth and Ross; and also on the right bank of the Wye, a little above Tintern Abbey. 2. Coarse-grained sandstone, of different colours-choco-late, brown, or reddish. These sandstones interstratify with clayey marls of a red or green colour. A quarry of them supplies flagstones. These are well-developed in Breconshire, at a place called the Daren, near Crickhowell, and at Manr: Bwlch y Chwyth, near Trecastle. II. CORNSTONES AND MARLS. This division consists of clayey marls, red and green and spotted, interstratified with bands of sandstone, and with irregular courses of what are called in Herefordshire, Cornstones. The cornstone is an impure, earthy limestone, consisting usually of small concretionary lumps only, but occasionally appearing in large subcrystalline masses. These cornstones are, in colour, mottled, red, or green. The cornstones and maris exhibit themselves in the Vale of Usk, near Abergavenny; in the Cliffs of Llanstephan Castle, below Carmarthen; and in the western face of the Brown Clee Hill, near Ludlow. III. TILESTONES. This lower division of the Old Red Sandstone consists of micaceous and quartzy grit, finely laminated or slaty, and is therefore easily separated into flags for causeways, and tiles for roofing houses. In consistency they are very hard, and in colour, reddish or green. These Tilestones can be most advantageously studied in a section under HOREB CHAPEL, in Cwm Dwr, a few miles to the south of Landovery, and in one of the heights above the River Teme, between Ludlow and Downton Castle. This cutline is sufficient to give a general notion of the lithological characteristics of the Old Red Sandstone. Its most magnificent and enormous development is in South Wales and Herefordshire; but immense beds of it are also found in Devonshire, and in the northern parts of Scotland. On the continent it is largely developed in the provinces on the Rhine; and in Russia, it alone covers an area larger than the entire space of Great Britain. In different countries it may be expected to differ in mineral character. In South Devonshire and in Cornwall, the Old Red Sandstone rocks are found associated with slaty and limy strata, which consist of green chlorite slate, alternating with quartzy shales and sandstones, accompanied with blue and grey limestones, which pass into red sandstones and conglomerates. This class of rocks was formerly referred to a group called "the transition," or the most ancient fossiliferous series, till they were ascertained by Sir R. Murchison and Professor Sedgwick to be really Old Red Sandstone beds, and which they demonstrated by their true position in the series, and by their type of fossils. In Scotland, the uppermost parts of the Old Red Sandstone differ from those of Herefordshire and South Wales. These parts are best seen at their junction with the lowest member of the coal-measures, as developed in Dura Den, to the south of Cupar, in Fifeshire. There it consists of— 1. Bands of yellow sandstone, in which the scales of a sauroid, or lizardish fish, called Holoptychius, and the fossil Pterichthys were first discovered. 2. The Sidlaw and Strathmore beds-which are seen between Stonehaven and the Frith of Clyde, forming the Sidlaw hills and the valley of Strathmore-a series of beds several thousand feet thick, and lying as follows: c. Red and mottled marls, cornstones and sandstones. II. THE ORGANIC REMAINS OF THE DEVONIAN. 1. The Old Red Sandstone, as we have said, used to be thought so destitute of fossils, that it was difficult to mark it as a separate group; but through the researches of Murchison, Sedgwick, and Hugh Miller, it has been raised to the rank of an independent formation of paleozoic rocks, distinguished by a well-defined type of organic remains. This was done by the discovery and by the establishment that the Devonshire limestones, which were so rich in fossils, were in reality rocks of the Old Red Sandstone age. 2. In the Old Red Sandstone we find distinct and new forms of existence-that is, new when compared with the inhabitants of the waters that had deposited the grey rocks of the Silurians beneath them; and distinct both from them, and also from the organisms of earliest bands of the coal-measures above them; so distinct, indeed, that they altogether differ from any animals found in either of the newer formations. 3. The fossils occur chiefly in those strata where limy matter is most diffused. Scales and some fragments of fish have been found in cornstones, but the best preserved fossils are supplied by the finely-laminated flagtones, and in the marls, which are contiguous to limestones. 4. PLANTS. For many years the only traces of plants found in the Devonians consisted of impressions of sea-weeds, or fucoids, which were discovered in great abundance on the surfaces of laminated sandstones in Forfarshire. Afterwards, Mr. HUGH MILLER discovered at Cromarty, in the lower division of the Old Red Sandstone, the remains of a LAND-plant. This deserves to be remembered, as it is, probably, the most ancient cone-bearing wood in the creation. Since then, land-plants have been found in the sandstone at Lerwick in Shetland, something like the calamites or horetails found in the coal-rocks. In the Devonians of Ireland, also, there have been found at Knocktopher, Kilkenny, species of tree-ferns differing widely from those of the overlying coal. This has been called Cyclopteris Hibernicus, and deserves also to be remembered, as being the oldest of tree-ferns as yet known in the crust of the globe. It has been lately proved by Dr. Flemming, that fruit-like clusters found in the paving-stones of Forfarshire, which were -supposed by Sir Charles Lyell to be the eggs of a shell-fish, and by Dr. Mantell to be the eggs of a batrachian, or frog-like animal, are, in truth, a receptacle of a land-plant covered with carpels or fruits, 5. ZOOPHYTES. Among the limestones of the Old Red Sandstones, Corals and Crinoidea, or lily-shaped corals, are very numerous, many of which are common to the mountain limestone, to the Silurian lime-rocks, and to the Devonian marbles, as may be instanced in the beautiful coralline marbles of Torquay and Babbacome Bay in Devonshire. The most abundant corals are the Favosites and the Cyathophyllum. The following engraving (fig. 27) represents the polished surface of a specimen of Favosites: Fig. 27. Favosites Polymorpha. 6. SHELLS. The Devonian beds, which were found in the sea and are called marine strata, are very rich in fossil shells, and contain some of peculiar types. In the Old Red Sand-stone of Scotland shells are rare, while that of Devonshire abounds in shells and crustaceans, or animals who, like the crab, live in a bony crust. Some of the most numerous shells are the Buccinum, Turbo, Pecten, Avicula, etc. In reference to the shells, one of the most remarkable circumstances is, the great abundance of some of the most ancient types that lived during the Silurian period-such as Atrypa, Spirifera, and Terebratula. Among the higher order of shells, we have the Bellerophon. The forms of shells which are exclusively Devonian are, the Calceola Sandalina, and the Strygocephalus megalodon, Cucullatus, and Clymenia. One of the most peculiar crustaceans of the Devonian is a species of trilobite, called Brontes Flabellifer. 7. FISHES. It is in the Old Red Sandstone that we first meet with an animal vertebrated, or having a back-bone. In the vast thickness of the Silurian rocks, no beds furnish any trace of a back-boned animal, except those which are the highest and latest, and which usher in the Devonian age. Though the Old Red Sandstones present their grandest development in Herefordshire and Breconshire, they, in the whole of that district, furnish but very few fragments of fish, while in Scotland they yield scarcely any other class of fossils. The entire Devonians of the globe have supplied 100 species of fish, and of that number sixty-five have been furnished by Scotland. The most prevalent of these remains are fossil fishes that belong to the genera Holoptychius, Pterichthys, Cephalaspis, Coccosteus, Dipterus, and Diplopterus. The Holopthychius is a kind of sauroid or lizard-like fish, some of which were thirty feet long, and their scales were three inches long and about two and a-half in breadth. It had plates to cover and defend its head. This has been found in upper divisions of the sandstone. The Pterichthys, a name which means the winged-fish, had wing-like appendages attached to each side of the head, as represented in fig. 28. covered. Fig. 28. The Pterichthys-the upper side. Of the Pterichthys, eight different species have been disUpon the first discovery of them, the wing-like or the oar-like blades attached to their heads were supposed by Mr. Hugh Miller to be paddles, like those of the turtle, but M. Agassiz has shown that they were mere weapons of defence. In times of danger they were extended, as represented in the figure; but at other times they lay close to the creature's side. The tail seems to have been the only instrument of motion in the waters. The species found in the British Devonians are generally small, from eight to ten inches long. Some, however, were much larger, for in the Murchison collection there is one whose wing-like blades are a foot in length. The Cephalaspis, which means the buckler-headed fish, is a genus of fish that has its head covered by a shield or buckler, fig. 29. The plates which cover and defend the head are united into one bony case, jointed something like the plates of the lobster. The entire body of the fish is covered with scales, but the scales of the head are highly ornamented with radiated markings. Many specimens of this fish have been found in England, Scotland, and Russia, but the discovery of them has been chiefly limited to the middle group of the Old Red-Sandstone called the cornstones and marls. The Old Red Sandstone has furnished specimens of other kinds of fish, called Dendrodus, Dipterus, Diplopterus, Osteolepis, Glyptolepis, Cheiracanthus, etc. The one thing in which they all agree is, that they have all bony or horny plates to defend the upper parts of the body and the head. The Coccosteus, or the berry-bone fish, had a large head | period. For instance: the eruption of the molten granite of covered with bony plates, like the Pterichthys. In its mouth Dartmoor would be likely to affect both the Devonian beds and it had conical teeth, and its body terminated in a long flexible the coal measures that were in contact with them, so that the tail, which was the instrument of motion. Four or five dif- clear order of the strata can hardly be ascertained. Some of the ferent species have been discovered, varying in length from a laminated, slaty, or schistose bands have been so metafew inches to two feet. morphosed or altered, and so mineralized, that they have every appearance of belonging to the oldest primary or Silurian rocks. The crystalline aspect, and the slaty cleavage of these Devonian rocks, were impressed upon them long after the age of their actual deposition, and that by the eruption of the Dartmoor granite. The change also in colour, from the red sand of the South Wales Devonian to the red and grey muds of the Devonshire rocks, is by no means sudden, but resembles the gradual change exhibited in Pembroke. And if you imagine that in the region now called Devon, there was, in the Old Red Sandstone age, a less diffusion of the oxide of iron, you can account why the Devonshire beds have less of the red and sandy character than those of South Wales have. Fig. 29. Cephalaspis Lyellii. IV. SOME GEOLOGICAL PHENOMENA OF THE 1. The Old Red Sandstones are deposits that have been accumulating, for innumerable ages, in the profound depths of the ocean, into which large rivers and powerful torrents, charged with the detritus of the lands over which they flowed, emptied themselves, and embedded the remains of plants and animals that existed during that period. 2. In examining the geological history of the Old Red Sandstone upward, we find the following facts. At the close of the Silurian period, great changes took place both in the mineral character and in the colour of the deposits; for the grey mud of the Silurians was succeeded by red sediments, which were deposited over large areas of those seas. The red colour is probably derived from the oxides of iron which had perhaps been imparted and diffused by the wear and tear of some underlying gneissic or granitic rocks. 3. The hundreds or thousands of centuries occupied in forming deposits of such enormous thickness as 10,000 feet, or nearly one mile in depth in South Wales, must have formed in other regions accumulations, varying in character from those of Herefordshire and Breconshire, but still being what geologists call equivalent beds. On this supposition, the vast series of beds in North Devon, though differing in lithological character, are found to occupy precisely the same position of strata, that were formed in the same geological time as the Old Red Sandstone of South Wales. 4. There are geological facts, clearly ascertained, that will account for such variations in equivalent rocks of the Devonian 5. The change from the Silurian to the Devonian age was evidently very gradual. In the lowest beds of the Old Red Sandstones, that is, in the beds which lie nearest to the underlying Silurians, it is only by the detection of the upper Ludlow fossils, in the lower strata, or the tilestones of the Devonians, and which are often of a reddish colour, that the limit between the two groups of rocks can be distinctly marked out. The similarity between the upper Silurians and the lower Devonians show how gradual was the mineral transitions from the older to the newer. 6. The physical and mineral changes which took place at the commencement of the Old Red Sandstone, were accompanied by the disappearance of certain sea organisms that characterised the Silurian epoch, and they ushered in a new and an advanced creation of animals, adapted to the altered conditions of the globe. In the Old Red Sandstone we have the seat and commencement of the earliest vertebrata, or backboned animals, that have been yet recognised. In some of the underlying bands we detect, indeed, a few small fishes, but in the true Devonians we find a profusion of larger fishes, with their back-bones but imperfectly bony, and with their forms differing exceedingly from any kind of fish found in the present ocean. In this age, also, we meet with well-ascertained land-plants, which are of much larger dimensions than the very rare ones found in the upper Silurians. Towards the close of the Old Red Sandstone period, we find that an air-breathing animal, now called the Telerpeton-the first reptile in creation-was formed, and lived amid groves of tree-ferns and of cone-bearing trees, among the roots of which it could nestle. 7. At Louisville, Kentucky, North America, there is a magnificent display of one of the limestone beds of the Devonian age. In aspect, it much resembles a modern coral reef. Imagine an extensive area, whose surface is all a red, gritty sandstone, occasionally overflowed with the waters of large river like the Ohio. When such a sandstone is acted upon by the impetuosity and the weight of the river, the softer parts of the surface rock are washed away. As this is done, new forms of the bed are developed. These are coral masses, which, being harder than the sandstone, resist the action of the water, and now stand out in relief, like small bushes, many of them sending out their branches from their erect stems, some of them five feet in diameter, and present themselves just as they were growing at the bottom of the Old Red Sandstone sea myriads of years ago. 8. The deep excavations or scoopings which are found in the beds of the Old Red Sandstone furnish clear examples of the enormous denudations which took place by the action of currents. In many districts the Silurians and the granitic rocks, which in our day form the surface, must have been, at one time, all covered with the beds of the Old Red Sandstone, but have since that time been washed away by the gradual action of running water. Of this enormous denudation, two examples will suffice. In Scotland, on the western coast of Ross-shire, the Old Red Sandstone forms three immense hills insulated from each other. These are called Suil Veinn, Coul Beg, and Coul More. The horizontal regularity of the beds of these three mountains prove that they were once continuous and formed but one mass. The proof is as clear as if a brick LESSONS IN ALGEBRA. wall had been broken through in two places, and had three LESSONS IN ALGEBRA.-No. LX. (Continued from page 659.) strations, it will be seen that they are precisely the same, But in geometry, AB is an expression for a single line, and There is no impropriety, however, in representing a geo- EBD = α, CBEC, ACB=d, the demonstration will stand thus: 1. By Euclid 29, 1, 2. And 3. Adding the two equations, 5. By Euclid, 13, 1, 6. Therefore ADC=h, a+c=g=b+ ď This notation is apparently more simple than the other; but it deprives us of what is of great importance in geometrical demonstrations, a continual and easy reference to the figure. To distinguish the two methods, capitals are generally used that which is properly algebraic. for that which is peculiar to geometry; and small letters, for APPLICATION OF ALGEBRA TO GEOMETRY. IT is often expedient to make use of algebraical notation, for expressing the relations of geometrical quantities, and to throw the several steps of a demonstration into the form of equations. By this, the nature of the reasoning is not altered. It is only translated into a different language. Signs are substi. If a line whose length is measured from a given point or tuted for words, but they are intended to convey the same meaning. A great part of the demonstrations in geometry really consist of a series of equations, though they may not be pre-line, be considered positive; a line proceeding in the opposite sented to us under the algebraic forms. Thus the proposition, direction must be considered negative. If A B, fig. 2, reckoned that the sum of the three angles of a triangle is equal to two right angles, may be demonstrated, either in common language, or by means of the signs used in algebra. Let the side AB, of the triangle A B C, fig. 1, be produced Fig. 2. D A from DE on the right, is positive; Ac on the left is negative. to D; let the line вE be drawn parallel to a c; and let GHI Hence, if, in the course of a calculation, the algebraical value be a right angle. The demonstration in words, is as follows: 1. The angle E B D is equal to the angle BA C. (Euclid 29, 1.) 4. If to these equals, we add the angle A B C, the angle C BD CBE ACB Euclid 29, 1 CBD+ABC= 2GHI 6. Therefore BAC+ACB+AEC=2G HI. By comparing, one by one, the steps of these two demon of a line is found to be negative, it must be measured in a direction opposite to that which, in the same process, has been considered positive. In algebraical calculations, there is frequent occasion for But how, it may be multiplication, division, involution, etc. asked, can geometrical quantities be multiplied into each other? One of the factors in multiplication is always to be considered as a number. The operation consists in repeating the multiplicand as many times as there are units in the multiplier. How then can a line, a surface, or a solid, become a multiplier? To explain this it will be necessary to observe, that whenever one geometrical quantity is multiplied into another, some particular length is to be considered the unit. It is immaterial. what this length is, provided it remains the same in different parts of the same calculation. It may be an inch, a foot, a rod or a mile. If, for instance, one of the lines be a foot long, and the other half a foot; the factors will be, one 12 inches, and the other 6, and the product will be 72 inches. Though it would be absurd to say that one line is to be repeated as often as another is long; yet there is no impropriety in saying that one is to be repeated as many times as there are feet or rods in the other. This, the nature of a calculation often requires. If the line which is to be the multiplier is only a part of |
