LESSONS IN ARCHITECTURE. - No. I. ARCHITECTURE is the art of planning, constructing, and adorning public or private buildings according to their intended use. The word architecture is derived from the Greek apxw, archo, I command, and rixтw", tecton, a workman. This etymology (derivation) indicates the operatives engaged in the building on the one hand, and the leader or chief, the man of science and practical skill, putting in action all his resources in order to execute his plan on the other. Such a division as this was, no doubt, established from the beginning of the art. According, therefore, to the literal meaning of the etymology, mankind must have, at the origin of architecture, possessed a degree of civilisation sufficient for the organisation of different kinds of industrial operations, and acquired a degree of skill in the art, which enabled some men by their experience to be the leaders or directors of others. In this way, we may suppose that the art itself, or rather the symmetry, the harmony of proportions, and good taste in structures, at first began to be developed. Before arriving at this point mankind must have overleapt ages. One of the first wants of society was a covering or shelter from the inclemency of the weather, whether of heat or of cold. Simple was the art employed in constructions of this kind. Grottoes or caves hollowed square to make them more habitable, and cottages constructed of branches of trees and blocks of stone-such were the primitive constructions in wood and stone which formed the rudiments of architecture. From the simplicity of early structures men passed to the study of proportions; they then dared to attempt the grand; and, at last, reached the sublime. The origin of architecture cannot be assigned to any particular country. Every nation produced its own art, or style, by employing the various materials within its reach, and by giving to them such forms as their wants required. Proceeding at first from the high table-lands of Asia, in order to people the earth, the early fathers of our race could have but little idea of architecture, or of a well-established system of construction. As wandering and pastoral tribes, they lived in tents or wretched cottages, which had no pretensions to architecture. It was not until they became more settled that they sought the means of rendering their buildings,more durable, by employing in their construction, wood or stone, and bricks baked in the sun. From the differences in the materials, and from the variety of tastes and feelings, arise the varied appearances which the monuments of different nations present, and which constitute their peculiar style of architecture. Thus the Egyptian, born in the hot climate of Africa, in a country destitute of wood fit for building, and near the mountains of the valley of the Nile, containing large blocks of freestone and granite, created for himself a vigorous style of buildings, which completely sheltered him from the burning rays of the sun. These buildings were formed of colossal masses, which were easily transported along the waters of that famous river. The Greek, inhabiting a milder climate, surrounded by forests and quarries, gave a lighter form to his edifices; and employed wood in their construction, which harmonised well with the marble-a material of which the fineness admitted of a greater delicacy of structure and arrangement. The Chinese, surrounded by rivers bordered with bamboo, had only a meagre and tortuous species of architecture, as ephemeral in its duration as it was fragile in its origin and construction. The very different character exhibited in local architecture enable us to judge of a country by its monuments, inasmuch as the buildings themselves are the expression of the various wants of the people who con VOL. I. | structed them. It is easy to understand how their different arrangements and structures are but the reflection of the religion or the manners of the people. The general style of the monuments of a country is a durable image of the different phases of its civilisation. In these, we see it in its primitive, refined, or degraded state, as civilisation arose, approached to perfection, or decayed. Nations naturally established great divisions in their architecture. They first built their private dwellings; then their public buildings: and these, in their numerous subdivisions, constituted civil architecture. Religion caused them to build temples and other edifices, attaching to them ideas of duty and moral obligation thus arose sacred architecture. The fortification of their frontiers, their towns, and their conquered countries, gave birth to military architecture. In this hasty sketch, we see how extensive is the series of buildings which cover the face of the globe, some of which belong to the first ages of its history, and others of which are being rediscovered in our own day. The study of these will be duly appreciated by the historian, the philosopher, the archæologist, and the artist, who, each with his own particular view, knows how to find a great lesson in these silent witnesses of past civilisation, as well as in those existing in full vigour around us. Architecture is founded upon three great principles which ought to be immutable: 1, the useful, without which states and private individuals would be led into superfluous and ruinous expenses; 2, the true, because it ought to express in all its varied forms the great principles of construction upon which it rests; 3, the beautiful, which is the end of all the arts depending upon design, and no less of architecture the most useful. On these principles, every style of architecture has the same value; and an artist should not curb his genius by confining himself to the study of one particular style. It is only the man of talent, to whom the construction of an edifice is entrusted, who can combine the different arrangements and forms, harmonise the various parts, and particularly express by plans, skilfully worked out, the disposition of the whole or of every part of the building. Upon these arrangements and plans rests the reputation of an architect, and science demands of him a well-grounded assurance of the good construction and durability of his work. Architecture is not an imitative art, like her sister arts, sculpture and painting. We see nothing in nature like our buildings as a whole; or rather nothing which could serve to guide us in its applications, or in the harmony of its lines. In this art, man has done everything himself. He has employed matter; he has invented forms and proportions to produce in the minds of his fellow-creatures ideas correlative of order, harmony, grandeur, richness, and durability. He has been enabled, by the force of art, to give, as it were, thought to matter, without being indebted for his ideas to any of the external forms of nature. Like the poet and the musician, the architect can transport the spectator into an ideal world, by creating forms and effects formerly unknown; but, very different from them in results, he renders his creations palpable, and gives them durability. Moreover, the useful, the true, and the beautiful, must be ever present to his view; and, however fruitful his imagination may be, he cannot emancipate it from science, the eternal basis of all the productions of his art. The architect is not, therefore, as the vulgar think- and, more unhappily still, as ignorant men who usurp the title thinksimply a head workman or common decorator of miserable constructions, a species of animal always to be found. He spends his youth in the study of his art, and of the splendid examples left on the face of the old world by ancient civilisation. In conjunction with these studies he makes himself master of the exact sciences, in order that he may execute his plans also makes himself acquainted with the physical sciences, in with precision, and study the nature of their construction. He order that he may become acquainted with the nature of the materials which he must some day employ, and be able to cal18 eulate their effects. In short, he devotes himself to practical presented the square form, if they did not give them this form experience, and to the working part of architecture, in order by manual labour. Stonehenge in England exhibits a number to render himself capable of executing public or private build-of square pillars supporting enormous architraves, the whole ings, and to make himself responsible for the stability of appearing to have constituted a large and well-constructed edifices entrusted to him either by private individuals or by edifice. These evidences of the first attempts of past civilisa the state. BUILDINGS OF UNHEWN STONES. After these preliminary remarks, we may shortly trace the progress of architecture amongst the different nations of antiquity, for the purpose of reaching our own times in chronological order. Before entering into details, we may point out the particular features which characterise the grand periods of the art, and the different systems in which its resources were developed, in order to satisfy the numerous demands of the civilisation in which it originated. Architecture, like all the productions of the human mind, presents at first only simple rudiments, quite in accordance with primitive manners. From the earliest ages we find three great divisions established amongst all nations: 1st, private buildings; 2nd, religious edifices; 3rd, military constructions. The first care of a people, as we remarked before, would be to construct individual dwellings; but being at first hunters and shepherds, they would be necessarily wanderers, and their dwellings would be tents constructed of the skins of animals, or cottages made of branches of trees. When they dwelt on the borders of rivers they would employ reeds; Asia and Egypt present us with examples of this kind. In some exceptional cases they dwelt in caverns, or in shallow excavations. The cottages were usually circular; piles of stones and earth, arranged in a circle, constituted their foundation. This form is found amongst all nations; that of the square, requiring more complicated combinations was not adopted at first. The simplicity of the first erections for religious purposes may be seen in the construction of the altars of early times. The first sacrifices, which the Bible and ancient tradition trace up to the creation, were made upon consecrated heaps of stones, which they collected upon high places. These first altars, called BETH-EL (the House of God), were erected in Chaldea, in Judea, and in Egypt. They were built, according to the Scriptures, of stones without cement, if the places where they were raised afforded proper materials. In other places they were constructed of turf and earth, where the plain country presented no solid materials. Such erections or mounds are found in Asia Minor and in India; at Heliopolis, celebrated for the worship of the sun, and the great Sidereal divinity of the Syrians. Lucian describes a throne or altar to the sun composed of four great stones arranged in the form of a table. At Ortosia, in Syria, there is an edifice of this kind raised in an open enclosure, and built of stones in a square form. Strabo relates, that travelling in Egypt, he saw his road covered with temples devoted to the god Mercury, which were composed of two unhewn stones, which supported a third. Artemidorus, quoted by Strabo, mentions that in Africa, near Carthage, the god Melkart (Moloch), or the Phoenician Hercules, the worship of which was brought from Tyre, was worshipped in a similar manner, three or four stones being placed one upon another. This simple manner of building applied to primitive altars, and to the sacred enclosures which surrounded them, after having been developed, as we have seen, in Asia and Africa, extended into Europe from the borders of the Black Sea and the Caucasus, where M. Dubois, of Neufchatel, saw a great number, even to the Atlantic Ocean and to the northern seas. Pausanias describes some of those in Argolis, and recent travellers have seen others in Greece. It is well known that they exist in France, in England, in Norway, and in Sweden, where all these works of early civilisation are known under the name of Celtic and Druidical monuments. America presents numerous examples of similar constructions, which show howrising nations exhibit the same analogies, as their arts are in the process of formation. tion are gradually and daily disappearing under the progress of those which are being developed around them. Thus Asia has lost the most of her ancient monuments, owing to the early state of her progress in the arts. Africa, for the same reason, presents as few examples, although they are mentioned by ancient authors. Greece and Italy, and their neighbouring islands, only exhibit examples of the same kind in places nearly deserted. The northern countries of Europe alone preserve some, because that civilisation was later there; and the history of their sudden and unexpected conquests extends only to a period of about two thousand years. In America the later civilisation of the Aztecs (1196) and the Mexicans caused the primitive monuments around them to disappear, by the development of their own. This process is perfectly analogous to that which took place first in Asia, then in Greece, in Africa, and in Italy, and which we see taking place in the western countries, where every day they are destroyed in order that the materials may be used for roads and private buildings. This simple and primitive style of architecture appears to Fig. 1. have been originally universal, if it was not simultaneous with the progress of civilisation, which marched from east to west; and has left monuments and edifices so varied, as to occasion them to be classified, and have names given to each class. These names are borrowed from the old Celtic tongue, or language of the Druids. Thus, class 1st was called Peulvans, or Men-hirs, which consisted of long stones, erect and isolated (standing singly) like obelisks. (Fig. 1.) 2nd, Cromlechs, which were great circles, elliptical or spiral, formed of huge stones slightly elevated. 3rd, Dolmens, consisting of large tables or platforms of stone, supported by several huge vertical stones. (Fig. 2.) 4th, Rows, or Uncovered Alleys, of upFig. 2. right stones, placed in rows like trees, and occupying a very considerable area, like those of the plain of Carnac in the department of Morbihan, or the province of Brittany in France. 5th, Covered Alleys, or long rows of parallel stones set upright, and supporting masses of stone placed horizontally so as to form a ceiling. (Fig. 3.) 6th, the mili Fig. 3. tary constructions of early times appear to have been mounds or artificial hills, at the summit of which there was a shallow excavation, of which the edges formed a rampart. It is certain that in countries, where hills naturally occurred, they fortified them in the same way as those which were raised by art. And these Fig. 4. natural fortifications are still to be seen in the neighbourhood of Athens and the Piræus; they were of immense service in the last war of independence. Mankind in a savage or wandering state having no instruments for raising the earth or digging ditches, made fortified enclosures with heaped stones, having a Simple as this system of building is, for it cannot yet be called architecture, we recognise the periods of its commencement, its progress, and its development. Thus the most ancient of these edifices, such as were erected by the most ignorant people, were built of enormous stones in the shape double slope. The entrances to these fortresses were defended which nature gave them. Moreover they selected those which by artificial hills, placed inside near the gates. (Fig. 4.) QUESTIONS ON THE PRECEDING LESSON. What is the etymology of the word Architecture? What are the two classes of persons which the etymology indicates? What was the simplest kind of Architecture first resorted to ? Whence did men proceed to establish them in different countries? What was the nature of the constructions they then employed? Describe the origin of civil, sacred, and military Architecture. education should he have? State the nature of the buildings of unhewn stones, and for what purposes they were first employed. Mention some of the countries where the Celtic and Druidical remains are found. State the different classes of these buildings; and mention some of the Celtic names of these classes. LESSONS IN LATIN.-No. XVII. By JOHN R. BEARD, D.D, THE LATIN VERB (continued). Now, by comparing these tables together, you may learn that o is the sign of the present tense; re, of the infinitive mood; i, of the perfeet tense; and um of the supine. In other words, by adding o to am, you form the present tense indicative mood first person singular; by adding bam to ama, you form the corresponding imperfect tense; by adding i to amar, you obtain the perfect tense; by adding re to ama, you get the infinitive mood; and finally, by adding tum to ama, you make the supine; thus: You thus see that there is a present stem, an imperfect stem, a perfect stem, an infinitive stem, and a supine stem. Of these, the imperfect and the infinitive are nearly the same. Properly speaking, the present stem in amo is the same as the imperfect and the infinitive, for the second a is a part of the root. Hence amo is a contraction for ama o. We have previously seen that a long (a) characterises the first conjugation to which amo belongs; also, that e long (e) characterises the second conjugation; and i long (i) the fourth. Hence only one class of verbs is characterised by a short vowel, and that is the class which bears the name of the third conjugation. This e short (e) however does not strictly belong to the verb, but is only a connecting vowel between two consonants in this conjugation, the essence of which is that its stem is consonantal or ends in a consonant. Thus, in the infinitive mood is introduced for the sake of sound between the stem and the ending of the infinitive; e.g., leg (e)re, for legre; in the same way leg(e)bam instead of legbam. But the other conjugations have vowel stems, as ama, doce, audi. The verbs of the third conjugation are called strong, and appear to be the most ancient. The verbs with vowel stems bear the name of weak, and are of later origin. Thus am or am er 3. am abo am andum The future of the third and fourth conjugations have the terminations am, ar, instead of bo. The subjunctive passive of the third and fourth conjugations ends in ar instead of er. II. From the stem of the infinitive in ā, ē, I, and in the third conjugation from the consonantal stem with the connecting vowelě; that is, from ama, doce, lege, and audi, are formed 1. The imperfect subjunctive, active and passive. 2. The imperative, active and passive. 8. The infinitive passive. Accordingly, 1. The supine in u. 2. The passive past participle. 3. The participle future active. Accordingly, 1. amatu of simplicity. It will be a good exercise for you to draw out I have here confined myself to amo and its parts for the sake the forms of the three other conjugations according to these examples. The forms of the verb not mentioned above are made by combination with the participles and parts of the verb esse, to be. Thus the perfect, pluperfect, and second future passive are formed by joining to the perfect passive participle sum, sim, eram, essem, ero, or fuero; e. g., amatus sum, amatus essem, amatus ero, &c. The infinitive future active is formed by adding esse to the participle future active as amaturum esse. The infinitive perfect passive is formed by joining esse with the participle perfect passive as amandum esse. GENERAL VIEW OF THE TENSE-ENDINGS, INDICATIVE, and Tenses. Indicative. Subjunctive. PASSIVE. Indicative. Subjunctive. rer 1st, em or 3rd & 4th, ar erim tus sum tus eram tus sim tus essem tus ero or fuero Frequently, in order to understand a formation, you will require to know how letters are related one to another. For Imperfect. bam instance, the supine of lego is lectum. Here the g seems to have disappeared. It is however represented by the c. instead of the hard legtum we have lectum. In rexi the perfect Perfect. of rego, the g seems to have disappeared. But it has its re. presentation in the c or k in xi, thus rexi, if written according to the sound (phonetically) would be regsi or recsi (reksi). The sibilant (8) is also introduced for the sake of euphony.. To pursue this subject in detail would require more space than we have to give. It must suffice to have put you in the right direction. When your ear by constant practice is accustomed to the combinations of letters which the Latins were fond of, you will have received a great assistance towards correctly forming the several parts for yourself. Let us now take up the chief parts separately, and the present stems, am(a), doce, leg, and audi. From these are formed If you take from the first person of any tense the termina- | jugations the tense-stem, as in the subjunctive present, ends in tions 0, 1, r, and where or appears the syllable or, then you get am, ar; as,the tense-stem which appears in all the other persons of the tense. This you may see exemplified in the ensuing legam, Active. legar, Passive, But the a of the first person is changed into e in the rest; as,— audiam, Active. audiar, Passive. amavi, amav, Perfect Indicative. amaveram, amavera, Pluperfect Indicative. amaverim, amaveri, Perfect Subjunctive. amavissem, amavisse, Pluperfect Subjunctive. So it is with the three other conjugations. To these stems are added the consonantal person-endings just given. If the tensestem ends in a vowel (except u) the person-endings are made without any connecting vowel; e. g., am doc II. But if the tense-stem ends in a consonant or in u, you must employ a connecting vowel; as shown thus: The course of instruction through which you have now gone will require constant repetition. When you have made yourself master of the forms which ensue, by imprinting them on your memory, you will do well to go over and over again these instructions. With diligence combined with observation you will then make yourself familiar with the Latin verb, not as a mere matter of rote, but understandingly; knowing well how the parts are formed one from the other, and how they are all connected with the common stem. I advise you, however, to question yourself very narrowly, and again and again, before you attempt to pass to the conjugation-forms which I am about to supply you with, though you will do well to refer to these forms for aid in understanding my remarks, and seeing their application. GENERAL VIEW OF THE TENSE AND PERSON ENDINGS OF ALL FOUR CONJUGATIONS. P. āmus, ātis, ant eo, es, et IMPERATIVE. ACTIVE. I. em, és, et ēmus, ētis, ent SS. P. emus, ētis, ent a or ato, 3 ato PRESENT. aud IV. IMPERFECT. amay I. docu II. S. i, isti, it ĕrim, eris, ĕrit leg III. (P. ĭmus, Istis, erunt ĕrimus, ĕritis, audiv IV. or ĕre [ĕrint amay I. S. ĕram, erat, erat issem, isses, isset docu II leg III. (P. ĕramus, ĕrātis, ĕrant issemus, issētis, audiv IV. amav I. docu II. S. ĕro, ĕris, erit leg III. audiv IV.) P. ĕrimus, ĕrītis, erint [issent LESSONS IN GEOMETRY.-No. VIII. LECTURES ON EUCLID. HAVING in our previous lessons on geometry, given our readers some kind of insight into the practical utility of the science of geometry, and having been urged by a numerous class of readers to give them something more elementary and initiatory, something more adapted to the young or to the old beginner, something that they really can understand on this most interesting subject, we have been induced, in consequence of the publication of the "Elements of Euclid" at such a price as to reach almost every class of purchasers, to commence à series of lectures on geometry in connexion with that masterpiece of antiquity. In these lectures we propose to smooth the way to the acquisition of this most important branch of human knowledge, by explaining in familiar terms the technicalities of the science at every step, even at the expense of repetition; and to show the practical application of each proposition in succession, as far as it can be done without introducing an amount of reference to branches of knowledge which the student cannot be supposed to have yet acquired. In carrying on these lectures we shall suppose ourselves addressing a class of students at a college, a people's college if you please, and we shall frequently use that colloquial style which serves to impress the mind and draw the attention pointedly to the subject in hand. One way of exciting the attention of the student will consist in amusing him with the different explanations, demonstrations, and solutions which have been given of various theorems and problems, and enabling him to judge of, or criticise if you will, each subject as it comes before him in the course of the lecture. Like all teachers who have the progress of their pupils at heart, we shall be glad to remove difficulties and give explanations when applied to by letter, for the benefit of all; but we hope that our correspondents will continue to be as they generally have been hitherto, courteous and civil in their manner of proposing their difficulties and requests, and that they will not be too positive as to the discovery of what they may fancy to be errors, lest they should turn out to be in error themselves. There is no science in which a man is more likely to be mistaken in his conceptions, especially at first, than the science of geometry, and we therefore desire to impress both ourselves and our readers with the necessity of an humble and teachable spirit, in the pursuit of this noble science. Two thousand one hundred and fifty years have rolled away since Euclid flourished at Alexandria, and had the king of Egypt for one of his pupils; and during all that period no greater genius has risen to publish a work on Geometry, which can take the place of his. Why is this so? Because he has adopted the strictest system of pure reasoning that the world ever saw in his whole system, and in his demonstrations; and, although his work is not without faults, we do not think it will ever be supplanted in Great Britain by any better system. You will, perhaps, ask me why in Great Britain? We answer, because the British mind is like the Greek mind. It is of a noble and solid nature, and the studies of ancient Greece have ever been cherished by the inhabitants of this island, owing to the striking similarity of their minds. The French, our amusing and volatile neighbours, have greatly abandoned Euclid. They have adopted systems of their own; ingenious and useful systems, and possessing marks of great originality; but they want the vigour and solidity of the Greek naster, the favourite of the British in Geometry. Even Professor de Morgan, that eminent analyst, and original but singularly obscure writer says, in his "Elements of Algebra," "In England, the Geometry studied is that of Euclid, and I hope it will never be any other; were it only for this reason, that so much has been written on Euclid, and all the difficulties of Geometry have so uniformly been considered with reference to the form in which they appear in Euclid, that Euclid is a better key to a great quantity of useful reading than any other." Though the British, as a nation, appear to surpass the French in the practical application of the principles of Geometry, and though we believe that this is owing to aeir being well grounded in these principles as laid down in Euclid, yet it cannot be denied that the French have long been our masters in the higher branches of Mathematics. Of late years, indeed, there have been some evidences of a healthy attempt at original writing on these subjects, and among these, Professor de Morgan deserves credit for the share he has had in this onward progless; but we cannot refrain from strongly recommending simplicity in writing and in notation to all our original writers. We have our own opinion, that by a more general spread of mathematical knowledge over the face of our happy island, some new lines of investigation and discovery may be struck out; and that, by the fortunate invention of some original genius, even Euclid himself, great as he is, may one day perhaps be thrown into the shade. There is the less chance, however, of this taking place when we consider the length of time that he has already maintained his ground; and when we find, upon examination that he has followed the most natural course in the study of geometry which the human mind could be supposed to follow. In fact, Euclid is simply the expression of the great human mind itself in one of its most interesting studies-the parent and the germ of the science of future ages. If Euclid had not done what he has done, some one else must have done it, and that some one else must have had the glory and the renown of the immortal work ascribed to his name. But it is now time to enter upon the consideration of the work itself. In the lessons which we are about to give, we shall, for the sake of explicitness, and the clear distinction between what is Euclid's and what is not,—at least, as given out by his celebrated editor, Dr. Robert Simson, of the University of Glosgow,-cause to be printed in Italics, all that is really considered to belong to his Elements of Geometry. To this we shall add explanations, demonstrations, annotations, and criticisms of various kinds, all tending to elucidate the propositions in hand, and to show their practical bearing in the sciences and in the arts. We shall also add exercises for solution in a limited time, and if these be well and elegantly done by any of our pupils, we shall have much pleasure in giving the solutions a place in our work, for the instruction and encouragement of others. The question of prizes for solutions has been suggested to us; this is a grave question, and one that will depend on the continued success of this great national attempt to instruct the masses in the principles of Geometry. Geometry, as laid down by Euclid, consists of several books, in which the properties of different geometrical figures are explained and demonstrated, in a series of short chapters called Propositions. To each book is prefixed a list of Definitions, which fixes at once, and for all future investigations, the precise meaning in which the technical terms or names of things is to be understood. From these definitions, no departure is ever allowed in Geometry; and it is to this fixity of meaning, that the superior excellence of geometrical reasoning, above all other kinds of reasoning, is to be chiefly attributed. The definitions, however, are not mere explanations of terms or words; they are, in general, something considerably more than this; for they contain the statement or announcement of some simple and obvious property of the thing defined a property which belongs to that thing and to no other. In this view, the definitions may be considered as a series of self-evident propositions; or such, that if they are not at first self-evident, they must be taken as possessing this property, and made the foundations of our reasoning accordingly. In order to illustrate this by an example, let us take Euclid's definition of a circle : “A circle is a plain figure contained [or bounded} by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another." Now what is this definition but a proposition in which a certain important property of the circle is involved? viz., that all straight lines drawn from the centre to the circumference are equal. This is the test of circularity, if we may use the term, the method by which we ascertain whether a given figure said to be a circle is really a circle or not. But it may be objected to this test, that no circle that ever was made by human hands, was absolutely perfect, or strictly agreed to the terms of the definition. To this we reply, that it is of no moment to the geometer whether this were ever the case or not; all that he requires for his purpose is that you will admit the existence of such a circle for the sake of argument, and then he is satisfied; then he can go on with his propositions and demonstrations, and raise you up a beautiful body of science on the strength of your admissions. Besides it is quite sufficient for the purposes, both of theory and practice, that we can make a circle so nearly true, as that no sensible difference can be discovered between the lengths of any two straight lines drawn from the centre to the circumference. Indeed, for the purposes of theory alone, this neat approximation to the truth is not at all necessary; for it is enouga that you admit, for the sake of argument, that any two straight |