Many attempts at calculating age of the Earth:

• Different cultures using religious texts/beliefs come up with different ages (thousands of years to hundreds of billions of years)
• Examples: counted up "Begats" and lifespans of Biblical figures (yields thousands of years)
• Early attempts by physicists and geologists predicted a few million to hundreds of millions of years
• Compared ocean salinity to known amount of salt in rivers; assuming fresh water proto-ocean, how long to salinate the seas? (tens of million years)
• Calculated current rates of sedimentation, count total thickness of sedimentary rocks, determine total age (a few billion years): standard for Geology at beginning of 20th Century
• Calculate cooling rates of molten iron, determined known surface temperature of Earth & its thermal gradients, assume no additional source of energy (90 million years or so maximum, possibly less): standard for Physics at beginning of 20th Century

Physical Stratigraphy
Sedimentary rocks, because they are deposited, naturally form horizontal layers (strata, singular stratum). Because of their layered form, strata allow geologists to determine relative time (that is, sequence of deposition of each layer, and thus the relative age of the fossils in each layer). These form the basic Principles of Stratigraphy. he first three principles were developed by Niels Stensen (better known as Nicholas Steno):

• Principle of Original Horizontality: because strata are deposited under gravity, they form horizontal layers
• If the strata are no longer horizontal, something has disturbed the sediments AFTER they became rocks.
• Can determine way up from geopetal (or "way up") indicators: surface sedimentary structures such as mudcracks, ripple marks, footprints, raindrop marks, etc.)
• Principle of Superposition: unless they have been disturbed, the strata at the bottom of a stack were deposited first, the ones on top of that are next oldest, and so on, with the youngest strata being the ones on top.
• Principle of Lateral Continuity: sediment extends laterally in all directions until it thins and pinches out or terminates against the edge of a depositional basin.

Richard Alley (PSU) gives some advice concerning geopetal indicators:

As Steno and others mapped out strata, they found that sometimes there were types of breaks (discontinuities) in the layers. These are called unconformities, and represent gaps in the rock record (periods of erosion and/or non-deposition). Hutton, of Uniformitarianism fame, studied these and recognized that they represented aspects of relative time.

Three main kinds of unconformities:

• Disconformity: Surface of erosion/non-deposition parallel with bedding planes
• Angular Unconformity: Surface of erosion which cuts across the bedding plane of lower strata, indicating these were tilted prior to erosion.
• Nonconformity: Erosional surface cut into crystalline (non-sedimentary) rocks

• Principle of Cross-cutting Relationships: any structure (fold, fault, weathering surface, igneous rock intrusion, etc.) which cuts across or otherwise deforms strata is necessarily younger than the rocks and structures it cuts across or deforms.
• Principle of Inclusions: any rock fragments included as sediment or xenoliths in a unit are from an older rock unit than the one in which they are included (really, a special case of cross-cutting)

(Here is a great blogpost about Siccar Point and Hutton's discoveries.)

Using these principles, early geologists were able to figure out the sequence of events of deposition, the changing local environments, and the folding, faulting, igneous intrusions, etc. for any particular section of rock. However, how could they extrapolate the sequence at one section with the sequence at another location, far away?