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The clue for 9-Down is erroneous: "Hot summer period named for the constellation Sirius." Sirius is a star, not a constellation.
This to the various commentators prompted by my earlier (erroneous) claim that the answer to clue to 51 Down regarding lines on a sphere was wrong: Actually good old Euclid provides the answer as to why only lines that run through a sphere's "poles" are straight lines. His definition of a "straight-line" is a line "which lies evenly with points on itself." I.e., arcs connecting two points on that line must be equal on both sides of the line. One can then see that this condition is only satisfied by great circles on a sphere, as only such lines would have an equal inclination of the sphere on both sides of the line, and thus equal arcs connecting two points on the line. Whew! And I hope this atones for my original mistake.
@SP I want to thank all the interlocutors above for the education in matters spherical. I'm brushing up my Greek by translating Euclid's "Elements," but haven't moved beyond Book 4, so I'm still stuck in, but also dazzled by, the intricacies of plane geometry. This exchange has whet my mind for the later books. Thanks again!
The answer to Clue 51 Down is incorrect: All straight lines on the surface of a sphere do not meet. E.g., the Equator and the Tropics of Cancer and Capricorn.
@Sebastian Sorry, to you and Hoosier, but the tropics are just as "straight" as the great circles that run through the poles. The clue does not state that the straight lines are equal to one another as the great circles you mention are. To repeat, the tropics are just as "straight" as your great circles, just smaller circles.
@SP SP, You're conflating the genus "line" with the species "straight-line." Isn't a parabola a particular kind of curved line?
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