A New Introduction to the Mathematicks: Being Essays on Vulgar and Decimal Arithmetick. Containing, Not Only the Practical Rules, But Also the Reasons and Demonstrations of Them; with So Much of the Theory, and of Universal Arithmetick Or Algebra, as are Necessary for the Better Understanding the Practice and Demonstrations. With a General Preface, Including a Panegyric, on the Usefulness of Mathematical Learning

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W. Johnston, 1758 - Algebra - 372 pages

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Page xi - ... might be able to transfer it .to other parts of knowledge, as they shall have occasion. For, in all sorts of reasoning, every single argument should be managed as a mathematical demonstration : the connexion and dependence of ideas should be followed, till the mind is brought to the source on which it bottoms, and observes the coherence all along, though in proofs of probability one such train is not enough to settle the judgment, as in demonstrative knowledge.
Page 300 - OPERATIONS WITH FRACTIONS A) To change a mixed number to an improper fraction, simply multiply the whole number by the denominator of the fraction and add the numerator.
Page xi - ... we may truly say nature gives us but the seeds of it; we are born to be, if we please, rational creatures, but it is use and exercise only that makes us so, and we are indeed so no farther than industry and application has carried us.
Page 272 - When first the marriage knot was tied Betwixt my wife and me, My age did hers as far exceed As three times three does three ; , But when ten years and half ten years We man and wife had been, Her age came up as near to mine As eight is to sixteen. Now tell me, I pray, What were our ages on the wedding-day...
Page xi - I said above, that the faculties of our souls are improved and made useful to us, just after the same manner as our bodies are. Would you have a man write or paint, dance or fence well, or perform any other manual operation dexterously and with ease? let him have ever so much vigour and activity, suppleness and address naturally, yet nobody expects this from him, unless he has been used to it, and has employed time and pains in fashioning and forming his hand, or outward parts, to these motions.
Page 270 - If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer. If the errors are unlike, divide the sum of the products by the sum of the errors, and the quotient will be the answer.
Page 75 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; and each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds ; and these into thirds, etc.
Page xi - ... this from him, unlefs he has been ufed to it, and has employed time and pains in fafhioning and forming his hand, or outward parts, to thefe motions. Juft fo it is in the mind ; would you have a man reafon well, you muft ufe him to it betimes, exercife his mind in obferving the connexion of ideas, and following them in train.
Page xviii - Phaenomena as these did come under the known Laws of Motion, it might very well be taken for granted, that the more obvious Appearances in the same Fabrick are owing to such Causes as are within the Reach of Geometrical Reasoning.
Page xxiii - ... nor was there ever any thing that has contributed to enlarge my apprehensions of the immense power of God, the magnificence of his creation, and his own transcendent grandeur, so much as the little portion of astronomy which I have been able to attain. And I would not only recommend it to young students, for the same purposes,, but I would persuade all mankind, if it were possible, to gain some degree of acquaintance with the vastness, the distances, and the motions of the planetary worlds, on...

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