70 Plan of the Locks at Cincinnati, Ohio. 



3. That they now increase. 



4. That from this point, the bases and hjpothenuse have all a 

 common difference, in this case 200. 



5. As n increases from 1 to m, the ratio of the perpendicular to 

 the hypothenuse increases, till it reach its maximum, namely a 

 ratio of equality, when m=n. 



6. This ratio afterwards decreases from equality. 



7. The legs of the isosceles right-angled triangle, cannot be ex- 

 pressed in this manner, not being rational numbers. 



8. Nor for the same reason can the triangle of 30°, at the. base, 

 be thus expressed, since, though the base is half the hypothenuse, 

 the perpendicular is not rational. 



9. While n is less than m, the sum of the base and hypothenuse 

 remains constant, and is equal to the co-versed sine of the angle con- 

 tained by these two sides. 



10. When n exceeds wz, the sum of the base and hypothenuse, 

 (the former being negative,) is still equal to the same constant quan- 

 tity, but is then, the versed sine of the same angle. 



Quebec, December 28, 1832. 



Art. VII. — Plan of the Locks at Cincinnati, Ohio ; by Darius 

 Lapham, Assistant Engineer. 



TO PROFESSOR SILLIMATiT. 



Dear Sir. — As there have, within a few years, been many ini- 

 provements in the construction of locks on canals, which have not 

 been noticed in any treatise on that subject, within the knowledge of 

 the writer, it has been thought that a concise description of the locks 

 which are now in an advanced stage of construction in the city of 

 Cincinnati, Ohio, accompanied with a drawing, would be acceptable 

 to the numerous readers of your valuable Journal. Should you think 

 that the following d^cription and plan of those locks, have sufficient 

 merit to justify their insertion in the American Journal of Science and 

 Arts, they are entirely at your disposal. 



From the intersection of the Miami canal with Main street, in the 

 city of Cincinnati, to the lowest water in the Ohio river, there is a 

 fall of one hundred and twelve feet, which is overcome by ten locks ; 

 nine of eleven feet, and the lower one, of thirteen feet lift. They 

 are placed in a right line, and at such distance apart as will admit 

 of two boats passing each other between them. For the purpose of 



