construct x =ab by using compass alone, if a and b are given segments. I found the problem in the book "What is mathematics?".

The following is a description of Mohr's constructions.(Macheroni problem)
  9) Find $x = ab$, if $a$ and $b$ are given segments.   

I found the question didn't give a segment denotes 1.I've no idea how to construct the x.  
Can anyone give me a hint?
 A: Ratios are naturally expressed geometrically in terms of height/width of rectangles (or equivalently by slope of the diagonal).
Notice how $x = ab$ is equivalent to
$$
\frac{x}{a} = \frac{b}{1},
$$
which suggests the following construction.  (As mentioned in the other answer, this all depends on choosing a unit length to represent the number $1$.)

A: The question makes no sense unless you have a $1$ given: $a$ and $b$ are lengths, so $ab$ is an area; the required $x$ is the length of a rectangle of equal area to $ab$ and of width $1$.
One construction is to plot axes, and mark $(0,a)$ on the $y$-axis and $(-1,0)$ on the $x$-axis. The line between them has gradient $a$, so prolong that to where it meets the (vertical) line perpendicular to the $x$-axis drawn through $(b-1,0)$. The intersection is at $(b-1, ab)$.
A: Its implied
I would assume from a cursory reading that finding a product implies some correspondence with a real number. Ergo, finding $x=ab$ implies that both $a$ and $b$ hold some sort of value, for which an $x$ can be found.  I can therefore assume that $a$ and $b$ have some sort of length measure associated with them; it makes no sense to me to take a product otherwise. Products are reserved for numerical values; there is no such thing as a purely Euclidean product.
Whenever you have definable lengths, you have a definable unit length, which forms the standard basis for the whole system.  A unitary segment may not be provided but that doesnt mean one isnt implied.
If a value for length is given then surely a unit is associated with it, even if it is simply the generic "units".
It really doesnt matter
It isnt the lengths that matter; its the proportions to a line segment we call unitary that matters.  Ratios are the key here, not some measurable length.
You can define your own unitary basis segment, from which segments $a$ and $b$ can be proportionally compared, and thus imparting to them a measure of your own if one isnt already provided.  This is what @SammyBlack has done in his answer. He has also constructed for you the product.
Euclidean vs Cartesian
Proportionality has meaning in Euclidean geometry; length does not.  Segments have length after you define the unit, which is what happens naturally on Cartesian plots, or rulers, or other such demarcated straightedges. Rulers and marking straightedges were never allowed in Euclidean geometry for a reason; its cheating the rules of the game. It imparts numbers where numbers don't belong. In fact superimposing a numbered grid, and therefore introducing length, to the Euclidean plane is one of Descartes greatest achievements; he invented graphing.
It wasnt that Euclid didnt have a concept of length or wasnt tempted to measure lengths with a ruler, its just that doing so was an "impure" construction.
Length isnt natural
They recognized even back then that what units of measure you use changes the amount you measure, and it was arbitrary and man-made. Artificial.  But there is a purity and beauty to lines and circles, uncontaminated by man and his subjectivity.
This is why it was, has, and continue to be important to reject the ruler or any notion of length, and to realize that length is a completely artificial construct.  Its easy to convince yourself that a segment has a length but every time you take a measure stick to it, youre using a non-Euclidean tool and defining a unit. That unit just happens to be on the stick rather than on the plane of the paper.
If the unit is arbitrary one must ask: what is lost, if anything, in the long run by repeatedly using a specific unit? Perhaps we fail to abstract well enough or to generalize well enough.
Units matter
Bare this in mind throughout all geometric constructions.  A segment may not change, but its length depends on the system of measure youre using - ie feet, meters, etc. are perfect examples - which you can completely arbitrarily decide by choosing your own standard unit.  A length is simply the quantity of this unitary segment required to find congruence with the segment being measured.
If, say, $a$ and $b$ are segments of 3 and 5 centimeters, the numerical product is easily seen to be 15. What doesnt seem very important at first is the unit for the lengths of these segments are implied to be 1 centimeter. Segments $a$ and $b$ share a ratio of 3:1 and 5:1 with this unit segment. A segment of this length may not be drawn but it is implied nonetheless.
If I however, without changing $a$ or $b$, expressed their lengths as 1.18 and 1.97 inches, their product becomes 2.325. What has changed is the unit, not the segments themselves.  What has changed is the relative size of the segment, drawn or not, that defines the unit. Since the unit has changed the ratios of $a$ and $b$ to that unit have changed.
Because the segments $a$ and $b$ have not changed, which is the correct product: 15 or 2.325? Neither or both? I say neither; segments cannot be multiplied together at all, only their lengths can, and lengths dont exist without a unit against which measure is defined.
A product construction is no more complicated than a proportionality construction. The implied unit segment is the third required segment in the proportionality construction.
Finally, when you change your unit, $u$, segment by some proportion $\gamma$, you change the product by proportion $\gamma^2$.
But what about using rectangles?
Sure, it only takes two segments arranged as adjacent sides of a rectangle to define an area, which, in some sense is a product. But there are two caveats to this. The first is that you've changed dimensionality; youve gone from two one-dimensional objects to a single two-dimensional object.
More importantly, and more to the point, you cannot convert that area into a corresponding line segment. One ugly consequence to this is to attempt to resolve the question: how do you construct the product of three or four or five segments without escaping the confines of the plane? How do you determine the product of two areas? Or even how do you compare any two products depicted as areas for their relative sizes in a consistent and reliable way that preserves both order on the segments and order on their products? That leads to the second caveat, which is that without a unit to define a basis for measure, neither the line segments nor the area of the rectangle have a quantifiable measure to extract.  It would require a quantity - ie numerical values, ie proportions to a unit - to resolve these quandaries.
You see, there is no bijection between the area of a rectangle and the length of a line segment without some sense of quantity. Only through a notion of quantity we can biject each to the real number line where such preservations are guaranteed and areas can be converted to corresponding lengths. I can easily draw that 15 square-centimeter rectangle as a 15 centimeter long line segment.  Having that numerical 15 to work with though is tantamount.
I can also easily compare the product of 3 and 5 with that of 4 and 4 and of 2 and 6, etc. Expressed as numberless line segments arranged like rectangles, anywhere in the plane with any orientation, it isnt so clear which is the greatest or least product. But expressed as either numbers or as corresponding line segments, they are easily comparable.
