weithin
far and wide
Across a great spatial distance or over a large area; visible or spread far and wide.
Der Leuchtturm ist bei klarem Wetter weithin zu sehen.
The lighthouse can be seen far and wide in clear weather.

far and wide / widely
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far and wide
Across a great spatial distance or over a large area; visible or spread far and wide.
Der Leuchtturm ist bei klarem Wetter weithin zu sehen.
The lighthouse can be seen far and wide in clear weather.

The adverb 'weithin' (far and wide/widely) is a compound of 'weit' (far) and 'hin' (toward/there). It describes an extent or effect that stretches over a great distance ('weit') in a direction ('hin').
Das Feuer war weithin zu riechen.
The fire could be smelled far and wide.
The meaning of 'weithin' (far and wide/widely) depends on the context. In combination with verbs of perception such as 'sehen' (to see) or 'hören' (to hear), it means spatially 'over a great distance.' In combination with words like 'bekannt' (known) or 'anerkannt' (recognized), it means figuratively 'generally' or 'among many people.'
Die Ruine auf dem Berg ist weithin sichtbar. (räumlich)
The ruins on the mountain are visible far and wide. (spatial)
In its spatial meaning, 'weithin' (far and wide) is often combined with adjectives or participles that describe sensory perception. Common collocations are 'weithin sichtbar' (visible far and wide), 'weithin hörbar' (audible far and wide), or 'weithin zu sehen' (to be seen far and wide).
Der Jubel der Fans war im ganzen Tal weithin hörbar.
The fans' cheering was audible far and wide throughout the valley.
In its figurative meaning, 'weithin' (widely) describes the degree to which information or an opinion is widespread. Typical combinations are 'weithin bekannt' (widely known), 'weithin verbreitet' (widespread), and 'weithin akzeptiert' (widely accepted).
Die Ansicht, dass Bildung wichtig ist, ist weithin verbreitet.
The view that education is important is widespread.
far / wide
'weit' and 'weithin' can both have to do with great distance or large extent. 'weit' is very general and can be used as an adjective or adverb: 'ein weiter Weg' (a long way), 'weit weg' (far away), 'weit besser' (far better). 'weithin' is an adverb and means more like 'across a large area or distance' or 'generally among many people'. For example, people say 'weithin sichtbar' (visible far and wide) or 'weithin bekannt' (widely known). Learners may confuse the words because they sound similar and both have to do with 'great distance'. As a practical rule: 'weit' often directly describes a distance or a comparison. 'weithin' describes that something is visible, audible, known, or spread over a large area.

Der Turm ist bei klarem Wetter weithin sichtbar.
The tower is visible far and wide in clear weather.
Bis zum nächsten Bahnhof ist es ziemlich weit.
It is quite far to the next train station.