When you consider that there really is no market for old reel type lawnmowers and further understanding that not only is the motor and most of the drive components missing from this one, the likelihood of finding any repair parts would be slim to none, I would estimate the value at somewhere around $50/ton at the local scrap yard.

I found this paper advertisement for sale on eBay. Looks very similar to the mower you have. No luck in finding any lawnmower for sale though:
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While cleaning out my in-law's garage attic, I came across a hand-push reel mower with the following letters stenciled on both sides of the handle: "The Genuine Philadelphia Lawn Mower Company". There is a patent date of 1913 stamped at the top of the handle. Do you know how I can obtain additional information on the mower and whether it has any antique value. The mower appears to be in fairly good shape, considering its age, and the reel and wheels turn freely. Thanks.

Does anyone have an opinion on the best way to sharpen the blades of a push lawn mower?

To find information and parts, contact early day farm equipment organizations, the one who put on an annual show. These shows are centered around antique tractors and stream engines, but every thing else is there, too: lawn mowers, washing machines, garden tractors, butter churns, toys, broom makers, etc. The members of these associations are very helpful and they have access to many repair/restore sources. You might be able to sell you partial mower to someone. Get a member of an association to display you machine for sale. Don't expect it to bring a big price because the motor and drive belts are missing. These guys, although friendly and helpful, are hobbists and are always looking for the lowest cost item to work on.

I have been to a number of these shows and can state that you do have a very early motorized reel mower and very few existist today.

Many Ace Hardware stores have blade (and not just knives) sharpening services.

The sharpening procedure is the same as for modern reel mowers. On an old machine such as this one, do not try to grind away all nicks on the reel because this removes too much metal. Grind only enough to touch up the reel and form an edge. Expect to have nicks left over. Small nicks will not interfere very much it its cutting action. This machine is destinied for display if ever restored to working condition and the blades do not have to be brand new perfection. The major cutter is the straight blade at the bottom. If too much metal has been removed from this blade, it may not hold an edge. It may have been zone hardened; where the cutting edge was quenched to a greater hardness than the rest of the blade. Usually, the bottom blade is easily replaced (if you can find another one).

It was probably driven by a single cylinder engine of 2 to 4 horsepower.

I have a 1951 jacobsen power unit with reel attachment.One of 1242 made.The reel attachment is one of 599 made.It ran before I stored it.I would like to know what it is worth