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Astra Heights (X)

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### Geometric Theorems, Diophantine Equations, And Arithmetic Functions

##### By: József Sándor

... +PP 2 +PP 3 = 2A(ABC)=a =h; where a and h denote the lengths of sides and heights in the triangle. 1. Now let A 1 B 1 C 1 be a new triangle construct... ...4) which, in view of (2) is integer only if is even, = 2 ( > 0). The heights of a P-triangle are given by h a =b; h a =b; h c = ab c ; (5) 42 t... ...P-triangle are given by h a =b; h a =b; h c = ab c ; (5) 42 therefore all heights are integers only if cjab, which, by (2) can be written as (m 2 + n... ...ich is not a P-triangle (i.e. excluding the triangle (3; 4; 5)), all other heights cannot be integers. (11) Indeed, (2y 1)x 2 = = yt gives (2y 1)x... ...w ABC be an isosceles triangle with AB = AC = b, BC = a. Assuming that the heightsAA 0 =x andBB 0 =y are integers (clearly, the third heightCC 0 =BB 0... ... tiint i ce, No.44, 1987, pp.1-18. 4. J. S andor, An irrational series, Astra Mat. 1(1990), 24-25. 5. A. Horv ath, On di erentiable functions pres... ...3, 1989, pp.1-10, Univ. Timi soara. 9. J. S andor, An irrational series, Astra Mat. (Sibiu), 1(1990), No.5, 24-25. 10. J. S andor, On the regular t...

### Aesthetics

...who committed an “ideological crime”, that is the ascension to “spiritual heights”. A “play” without actors, scenery and dialogue consists in the re... ..., no.1-2, p.53. Brumaru, Al.I. , Interview with Octavian N. Voinoiu, in “Astra”, Brasov, no.244-246, 1996, p.75. Vlad, Mihail I., The Florentin S... ...96, p. 323,457, 494. Brenciu, Mircea, The Aesthetics of Paradoxism, in “Astra”, Brasov, no.7-12, 2nd year, April 1997, p.131. Carstoiu, Ada, Fl... ...n exceptional analyze of the Romanians’ alienation under totalitarism, in “Astra”, Brasov, 1997, pp.91-92. Ichim, Dumitru, The paradoxology of life...

### Begin the Adventure : How to Break the Light Barrier

##### By: Florentin Smarandache

...g of Roll Landauer of the IBM Thomas J.Watson Research Center in Yorktown Heights, New York, in which he concluded that there is no minimum energy re... ...l?id=americanR-motta- proceed\ingsOTSIC&_requestid=313. [7] Editors, Ad Astra journal, An Online Project for the Romanian Scientific Community, ht... ...l, An Online Project for the Romanian Scientific Community, http://www.ad-astra.ro/whoswho/view_profile.php?user_id=91&lang=en. [8] Gilbert, John,...

### Essays

##### By: Michel Eyquem de Montaigne

...s bow, his string, his arrow and his motion accordingly. Our counsels goe astray because they are not rightly addressed, and have no fixed end. No w... ...cheth us. ------ speculataque longe Deprendit tacitis dominantia legibus astra. Et totum alterna mundum ratione moveri, Fatorumque vices certis di... ...e they thought otherwise, and take them as tempests which shamefully lead astray the soule from her rest and tranquillitie? Vt maris tranquillita in... ... so strong. Nay, some there are that call scarcely think or heare of such heights. Let a beame or planke be laid acrosse from one of those two steep... ...de to sing. And another: Adde Heliconiadum comites, quoram unus Homerus, Astra potitus -- LUCR. 1. iii. 1081. Muses companions adde to these, of ... ...h the choise of one and other condition. For a man doth not fall from all heights; divers there are whence a man may descend without falling. Verily...

### The Life of John Sterling

##### By: Thomas Carlyle

...fter I last saw its mouldering rough- ness, I never read of fortresses, or heights, or spectres, or banditti, without connecting them with the one rui... ...hemists professed to do with organic substances in general, distil you an “Astral Spirit” from the ashes, which was the very image of the old burnt ar... ...there. But, on the whole, do not think you can, by logical alchemy, distil astral spirits from them; or if you could, that said astral spirits, or def... ...ot sun. An interesting Isle: a place of rugged chasms, precipitous gnarled heights, and the most fruitful hol- lows; shaggy everywhere with luxuriant ... ...t an easy matter (according to Sir Kenelm Digby), this of getting up your “astral spirit” of a thing, and setting it in action, when the thing itself ... ...s, but will certainly get me into worse.... But one must work; sic itur ad astra,—and the astra are always there to befriend one, at least as asterisk... ... were among his first inducements to settle in this new place. Open cheery heights, rather bare of wood: fresh southwestern breezes; a brisk laughing ... ...f the Beautiful; and not for a Priest at all? Striving to- wards the sunny heights, out of such a level and through such an element as ours in these d...

### Sartor Resartus: The Life and Opinions of Herr Teufelsdrockh

##### By: Thomas Carlyle

...an Editor enough to do. Ever higher and 51 Thomas Carlyle dizzier are the heights he leads us to; more piercing, all- comprehending, all-confounding ... ...his I know, If what thou namest Happiness be our true aim, then are we all astray. With Stupidity and sound Diges- tion man may front much. But what, ... ... the Donau bring down his mould-cargoes from the Carinthian and Carpathian Heights, and spread them out here into the softest, richest level,—intend t... ...rinding in the mill of Industry; one hunter-like climbing the giddy Alpine heights of Science; one madly dashed in pieces on the rocks of Strife, in w... ...Leather Breeches, and between the knees these memorable words, sic itur ad astra. Was not this the martyr prison-speech of a Tailor sighing indeed in ...

### Chronicles of the Canongate

##### By: Sir Walter Scott

...CTORY CHAPTER I MR. CHRYSTAL CROFTANGRY’S ACCOUNT OF HIMSELF . Sic itur ad astra “This is the path to heaven.” Such is the ancient motto at- tached to... ...n the one side from the banks of the Awe, and on the other wound round the heights of the mountain, with such a degree of accommodation to hill and le... ...had neglected to occupy, with directions to advance silently, and gain the heights above and in front of the hilly ground where the men of Lorn were c... ...the utmost fury, Sir James Douglas and his party shouted suddenly upon the heights in their front, showering down their arrows upon them; and, when th...

### Memorials and Other Papers

##### By: Thomas de Quincey

...ry, or at least to distinction—sic itur ad 24 Memorials, and Other Papers astra; unfortunately, it was not the way to Dublin. Conse- quently, on ever... ...s such, and to have commiserated his be- nighted brother, however wilfully astray, and however hate- fully seeking to quench that light for other men,... ...issaries drunk, cut the whole of them to pieces; and then encamping on the heights of Navarin, his lordship blockaded that fortress. The abruptness of... ...vals slowly agitated in the currents which now and then swept these aerial heights. In the centre of the lat- tice opened a single gate, on each side ...

### Sartor Resartus the Life and Opinions of Herr Teufelsdr Ockh

##### By: Thomas Carlyle

...or inspired, gives an Ed itor enough to do. Ever higher and dizzier are the heights he leads us to; more piercing, all comprehending, all confounding... ... this I know, If what thou namest Happiness be our true aim, then are we all astray. With Stupidity and sound Digestion man may front much. But what, ... ...de the Donau bring down his mould cargoes from the Carinthian and Carpathian Heights, and spread them out here into the softest, richest level,—intend... ... grinding in the mill of Industry; one hunter like climbing the giddy Alpine heights of Science; one madly dashed in pieces on the rocks of Strife, 1... ...f Leather Breeches, and between the knees these memorable words, SIC ITUR AD ASTRA. Was not this the martyr prison speech of a Tailor sighing indeed i...

### The Divine Comedy of Dante

##### By: Alighieri, Dante, 1265-1321

...ow was the sun so station’d, as when first His early radiance quivers on the heights, Where stream’d his Maker’s blood, while Libra hangs Above Hesper... ...: Credidit ut Cimabos picturae castra tenere, Sic tenuit vivens: nunc tenet astra poli. v. 95. The cry is Giotto’s.] In Giotto we have a proof at ho...

### The Divine Comedy of Dante

##### By: H. F. Cary

... CANTO I In the midway of this our mortal life, I found me in a gloomy wood, astray Gone from the path direct: and e’en to tell It were no easy task,... ...th the sound of trumpets, now of bells, Tabors, or signals made from castled heights, And with inventions multiform, our own, Or introduc’d from forei... ...ow was the sun so station’d, as when first His early radiance quivers on the heights, The Divine Comedy of Dante Purgatory 202 Where stream’d his ... ...: Credidit ut Cimabos picturae castra tenere, Sic tenuit vivens: nunc tenet astra poli. v. 95. The cry is Giotto’s.] In Giotto we have a proof at ho... ...lm, Where, on the gulf by stormy Eurus lash’d, Betwixt Pelorus and Pachynian heights, The beautiful Trinacria lies in gloom (Not through Typhaeus, but... ...y ones, that trac’d HIS steps, Whose marvellous life deservedly were sung In heights empyreal, through Honorius’ hand A second crown, to deck their Gu... ...vey l O beg ye grace for those, that are on earth All after ill example gone astray. War once had for its instrument the sword: The Divine Comedy of ...

...Excerpt: CANTO I. In the midway of this our mortal life, I found me in a gloomy wood, astray Gone from the path direct: and e?en to tell It were no easy task, how savage wild That forest, how robust and rough its growth, Which to remember only, my dismay Renews, in bitterness not far from death. Yet to discour...

### The Days Work

##### By: Rudyard Kipling

... many silver pieces of his proper value. Neither running water nor extreme heights made him afraid; and, as an ex-serang, he knew how to hold authorit... ...apped in a poshteen—a silk-embroidered sheepskin jacket trimmed with rough astra- khan—looked out with moist eyes and nostrils that dilated joyously. ...

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