The following GEOMETRICAL PROBLEMS may be solved by simple applications of the Propositions in the First Two Books of Euclid. If the student prefer going on to the Third Book at once, he may do so ; but he will find his progress quicker, if he from time to time practises himself in these or any other Geometrical Problems. Ir a perpendicular be drawn bisecting a given straight line, any point in this perpendicular is at equal distances, and any point without the perpendicular is at unequal distances from the extremities of the line. Any side of a triangle is greater than the difference between the other two sides. The difference of the angle at the base of any triangle is double the angle contained by a line drawn from the vertex perpendicular to the base, and another bisecting the angle at the vertex. To find a point within a triangle which is equidistant from the three angles. To describe a square which shall be equal to the difference of two squares, whose sides are given. To describe a rectangular parallelogram which shall be equal to a given square, and have its adjacent sides together equal to a given line. To describe a square which shall be equal to the sum of any number of given squares. To inscribe a square in a given right-angled isosceles triangle. To inscribe a square in a given quadrant of a circle. If straight lines be drawn from the angles of a triangle through any point, either within or without the triangle, to meet the sides, and the lines joining these points of intersection and the sides of the triangle be produced to meet; the three points of concourse will be in the same straight line. If the opposite sides or opposite angles of a quadrilateral figure be equal, the figure will be a parallelogram. The two triangles, formed by drawing straight lines from any point within a parallelogram to the extremities of two opposite sides, are together half of the parallelogram. If two sides of a trapezium be parallel, its area is equal to half that of a parallelogram whose base is the sum of those two sides, and altitude the perpendicular distance between them. If from one of the acute angles of a right-angled triangle, a line be drawn to the opposite side; the squares of that side, and the line so drawn, are together equal to the squares of the segment adjacent to the right angle and of the hypo tenuse. In any triangle, if a line be drawn from the vertex at right angles to the base; the difference of the squares of the sides is equal to the difference of the squares of the segments of the base. In any triangle, if a line be drawn from the vertex bisecting the base; the sum of the squares of the two sides of the triangle is double the sum of the squares of the bisecting line and of half the base. 120 THE ELEMENTS OF EUCLID. Given one angle, a side adjacent to it, and the difference of the other two sides; to construct the triangle. Given one angle, a side opposite to it, and the difference of the other two sides; to construct the triangle. If the three sides of a triangle be bisected, the perpendiculars drawn to the sides at the three points of bisection, will meet in the same point To trisect a given triangle from a given point within it. To determine a point within a given triangle from which lines drawn to the several angles will divide the triangle into three equal parts. Of all triangles having the same vertical angle, and whose bases pass through a given point, the least is that whose base is bisected in the given point. The sum of the sides of an isosceles triangle is less than the sum of the sides of any other triangle on the same base and between the same parallels. If from the extremity of the base of an isosceles triangle, a line equal to one of the sides be drawn to meet the opposite side; the angle formed by this line and the base produced, is equal to three times either of the equal angles of the triangle. THE END. LONDON: LONDON: 39, PATERNOSTER ROW: A Select Catalogue of BOOKS ON EDUCATION, FRINTED FOR LONGMAN, BROWN, GREEN, & LONGMANS. ELEMENTARY CREEK WORKS. Yonge: An English-Greek Lexicon; Containing all the Greek Words used by Writers of good authority; citing the Authorities in Chronological Order for every Word used; explaining the Construction; and giving the Declension or Conjugation of each word when irregular; and marking the Quantities of all doubtful Syllables. By C. D. YONGE, B.A. Post 4to. 21s. cloth. "This Lexicon is compiled on a most admirable plan, and will be found a most important, we will add an indispensable, assistant to the student seeking to perfect himself in Greek composition. Mr. Yonge furnishes a complete English vocabulary so far as there are equivalent and equipollent words in Greek to render the English term. The authorities are invariably given; and the lexicographer has displayed both taste and judgment, with infinite zeal to boot, in the selection of illustrative quotations. It is decidedly a work of rare merit." Church and State Gazette. Brasse's Greek Gradus. A Greek Gradus; or, a Greek, Latin, and English Prosodial Lexicon: containing the Interpretation, in Latin and English, of all words which occur in the Greek Poets, from the Earliest Period to the time of Ptolemy Philadelphus. By the late Rev. Dr. BRASSE. With a Synopsis of the Greek Metres, by the Rev. J. R. Major, D.D. New Edit. revised by the Rev. F. E. J. Valpy, M.A. 8vo. 158. cloth. Giles's Greek and English Lexicon. A Lexicon of the Greek Language, for the use of Colleges and Schools; containing -1. A Greek-English Lexicon, combining the advantages of an Alphabetical and Derivative Arrangement; 2. A copious English-Greek Lexicon. By the Rev. J. A. GILES, LL.D. New Edition. 8vo. 21s. cloth. ** The English-Greek Lexicon, separately. 7s. 6d. cloth. Dr. Kennedy's Greek Grammar. Græcæ Grammaticæ Institutio Prima. Rudimentis Etonensibus quantulum potuit immutatis Syntaxin de suo addidit B. H. KENNEDY, S.T.P. Edition. 12mo. 4s. 6d. cloth. Kühner's Elementary Greek Grammar. New An Elementary Grammar of the Greek Language. By Dr. Raphael Kühner, Co-Rector of the Lyceum at Hanover. Translated by J. H. Millard, St. John's College, Cambridge. 8vo. 9s. cloth. Valpy's Greek Grammar. The Elements of Greek Grammar: with Notes. By R. VALPY, D.D. New Edit. 8vo. 6s. 6d. boards; bound, 7s. 6d. Pycroft's Greek Grammar Practice. Three Parts: 1. Lessons in Vocabulary, Nouns, Adjectives, and Verbs in Grammatical order; 2. Greek, made out of each column for translation; 3. English for re-translation. By the Rev. J. PYCROFT, B.A. 12mo. 38. 6d. cl. Moody's Eton Greek Grammar in English. The New Eton Greek Grammar; with the Marks of Accent, and the Quantity of the Penult: containing the Eton Greek Grammar in English, and the Syntax By the Rev. and Prosody as used at Eton; with numerous Additions. CLEMENT MOODY, A.M. New Edition. 12m. is. cloth. Valpy's Greek Delectus, and Key. Delectus Sententiarum Græcarum, ad usum Tironum accommodatus: eum Notulis et Lexico. Auctore R. VALPY, D.D. Editio Nova, eademque aucta et emendata. 12mo 4s. cloth. KEY to the above, being a Literal Translation into English. 12mo. 2s. 6d. sewed. MESSRS. LONGMAN AND Co.'s Valpy's Second Greek Delectus. Second Greek Delectus; or, New Analecta Minora: intended to be read in Schools between Dr. Valpy's Greek Delectus and the Third Greek Delectus: with English Notes, and a copious Greek and English Lexicon. By the Rev. F. E. J. VALPY, M.A. New Edition. 8vo. 9s. 6d. bound. Valpy's Third Greek Delectus. The Third Greek Delectus; or, New Analecta Majora: with English Notes. In Two Parts. By the Rev. F. E. J. VALPY, M.A. 8vo 15s. 6d. bound. PART 1. PROSE. ** The Parts may be had separately. 8vo. 8s. 6d. bound. - PART 2. POETRY. Valpy's Greek Exercises, and Key. 8vo. 9s. 6d. bound. Greek Exercises; being an Introduction to Greek Composition, leading the student from the Elements of Grammar to the higher parts of Syntax By the Rev. F. E. J. VALPY, M.A. New Edition. 12mo. 6s. 6d. cloth. KEY, 12mo. 3s. 6d. sewed. Neilson's Greek Exercises, and Key. Greek Exercises, in Syntax, Ellipsis, Dialects, Prosody, and Metaphrasis. To which is prefixed, a concise but comprehensive Syntax; with Observations on some Idioms of the Greek Language. By the Rev. W. NEILSON, D.D. New Edition. 8vo. 5s. boards.-KEY, 3s. boards. Howard's Introductory Greek Exercises, and Key. Introductory Greek Exercises to those of Huntingford, Dunbar, Neilson, and others; arranged under Models, to assist the learner. By N. HOWARD. New Edition. 12mo. 5s. 6d. cloth.-KEY, 12mo. 2s. 6d. cloth. Donaldson's Theatre of the Greeks. The Theatre of the Greeks; or, a Series of Papers relating to the History and Criticism of the Greek Drama. Sixth Edition, revised and improved. With an Original Introduction and Notes by JOHN WILLIAM DONALDSON, B.D. Head Master of King Edward's School, Bury St. Edmunds; and formerly Fellow of Trinity College, Cambridge. 8vo. with Frontispiece and Wood Engravings, 15s. cloth. Dr. Major's Guide to the Greek Tragedians. A Guide to the Reading of the Greek Tragedians; being a series of articles on the Greek Drama, Greek Metres, and Canons of Criticism. Collected and arranged by the Rev. J. R. MAJOR, D.D. New Edition, enlarged. 8vo. 9s. cloth. Viger on the Greek Idioms. Translated and abridged, with original English Notes, by the Rev. J. SEAGER, Editor and Translator of " Bos on the Greek Ellipsis," "Hermann's Doctrine of Metres," "Hoogeveen on the Greek Particles," and "Maittaire on the Greek Dialects." 8vo. 9s. 6d. Post 8vo. 9s. bound. "The plan adopted in this gradus is a pleasing proof of the strides which true scholarship has made in the inferior grades of classical study. Instead of the old gradus, which seemed expressly constructed for the purpose of teaching the pupil to string words together with respect to the number and quantity of their syllables, without the least regard to sense or fitness, the present manual of metrical reference has the character of a critical apparatus which, while it supplies him with copious examples from the best authors, forces him at the same time to pay attention to the sense, and guides him moreover in the selection of such terms only as appertain to the age of pure Latinity." John Bull. |