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Clairaut's classic treatise on geometry remains a foundational text in the field to this day. Providing a clear and accessible introduction to basic principles like points, lines, angles, and curves, Clairaut's work has been an essential resource for students of mathematics for centuries. This translation by J. Kaines brings Clairaut's brilliant insights to a wider audience, making it an invaluable addition to any library. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.